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TT&5C 



TsH 



{English Translation with Mathematical 

Explanations & Notes) 

vol. II 




Present second volume 
translates all the verses in 
English. Translation is not 
literal but in mathematical 
terms, but preserving the 
technical terms in Sanskrit. 
Verses in praise of god have 
been left out, not because of 
disrespect. With all devotion 
inspired by Samanta 
Chandrashekhar, this is not 
the purpose of the second 
volume. In addition to trans- 
lation, each formula has been 
explained or derived accord- 
ing to modern mathematics 
and astronomy. The methods 
have been compared with 
other Indian astronomers and 
some times with other 
countries and with modern 
astronomy. This was the 
method and purpose, of 
Samanta himself. 

Technical terms and 
their calculations cannot be 
explained in words alone. So 
a general mathematical and 
technical introduction is 
given at beginning of each 
chapter with bibliography or 
source reference for further 
study. In that light only, the 
methods proved in the chap- 
ter can be understood. 
Where-ever considered use- 
ful, methods have also been 
explained with examples, 
based on text as well as 
modern astronomy. 




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SIDDHANTA-DARPANA 

(1899 A.D.) 

English Translation with Mathematical 
Explainations and Notes 



Vol. II 



Arun Kumar Upadhyay, IPS 

M.Sc, AIFC 




NAG PUBLISHERS 

11A/U.A., Jawahar Nagar, Delhi-1 10007 



This publication has been published with the 
financial assistance by Rashtriya Sanskrit Sansthan, 



New Delhi. 

Nag Publishers 

(i) 11A/U.A. (Post Offio 
Nagar, Delhi 110007. 



(ii) Sanskrit Bhawan, 12, 15, Sanskrit Nagar 

Plot No. 3, Sector-14, Rohini, New Delhi-85 
(Hi) Jalalpur Mafi, Chunar, Dist. Mirzapur, U.P. 

© Aran Kumar Upadhyay, IPS 

B-9, CB-9, Cantonment Road, Cuttack-753001 



ISBN 81-7081-342-9 (Set) 
ISBN 81-7081-406-1 (Vol II) 



Price : Rs. 



First Edition : 1998 



PRINTED IN INDIA 
Published by Shri Surendra Pratap for Nag Publishers, 

X^T^T^^S^ Bagh, 
Dayabasti, Delhi - 110035 



SUBJECT INDEX 

Chapter Verses No. Subject Pages 

No. 

INTRODUCTION 1-25 

1. Arrangement of book 1 

2. Numeration 2 

3. Transliteration of Sanskrta letters 5 

4. Survey of Indian Astronomy 6 

5. Astronomers of Orissa ' 12 

6. Candrasekhara 17 

7. References 24 

A. MADHYAMADHIKARA 

1. MEASUREMENT OF TIME 26-45 

1A Introduction 

1. Units of Measurement 26 

2. Units of length 27 

3. Measurement of time 31 

4. Accurate measurements 34 

5. Angles 36 

6. References 38 
1. TEXT 

1-22 Importance of jyotisa etc. 38 

23-39 Units of time 40 

40-56 Current time, angle units 43 

Units of length and time in 7 pages 



w Siddhanta Darpana 

2. REVOLUTION OF PLANETS 46-58 
2A. Introduction 45 

1-26 Translation of Text 53 

3. MEAN PLANETS 59.77 
1-13 Ahargana 59 
14-20 Lords of days etc. 62 
21-23 Calculation of Mean position 64 
24-46 Guru varsa 66 
47-51 Other methods 71 
52-70 Dhruva positions 73 
71-78 Hara, end 76 

4. CALCULATION AT DIFFERENT 78-112 

PLACES 

4A. Introduction and references 78 

1-21 Sphuta paridhi, palabha, desantara 88 

22-25 Cara correction 97 

26-32 Bhujantara correction 101 

33-41 Udayantara 105 

42-58 Padaka, dhruva, end 110 

B. SPHUTADHIKARA 

5. TRUE PLANETS 113-249 
5A. Introduction 

1. Concepts of planetary motion 113 

2. Modern Calculations 115 

3. Tables of sun, planets 137 

4. Equation for other planets 153 

5. References 155 
TEXT 

1-42 Nature of planetary motion, types 156 

43-75 Sara and its calculation 167 



Subject Index v 

76-89 Parocca of mangala, budha, sani 189 

90-112 Sighra and manda paridhi 195 

113-120 Bhuja, Koti phala 207 

121-123 True position of star planets 209 

124-131 Special methods for mangala, budha 214 

132-138 True sun, moon speeds 217 

139-158 True speed of tara graha 219 

159-160 Vakri position 225 

161-165 Heliacal setting 229 

166 Mean from true planet 232 

167-188 Use of Tables 232 

189-193 Udaya and asta times 236 

194-206 Calendar elements 238 

207-212 Extra and lost months etc. - 246 

6. CORRECTIONS TO MOON 250-379 

6A. Introduction 

1. Elliptical orbit of moon 250 

2. Deviations due to sun 252 

3. Corrections in defferent texts 254 

4. Modern charts of moon 259 

5. Indian charts 260 

6. Making a calendar 261 

7. Solar calendars in history 271 

8. Luni-solar calendars 276 

9. Old Indian calendars 281 

10. Indian eras 293 

11. Festivals and yogas 305 

12. References • 318 
TEXT 

1-16 Extra 3 corrections to moon 319 



336 
348 



Siddhanta Darpana 

17-26 Accurate motion of moon 322 

27-46 Need for accuracy, lambana 326 

47- 57 Origin from Bhaskara, smrtis 329 

58-67 Unequal naksatras 332 

68-74 Sankranti, different orbits 335 
75-91 Ayanamsa 
92-101 Kranti 

102-112 Variation in day length 352 

113-117 Udayantara calculation 35g 

118-120 Rising time of planets 359 

121-130 Equator rising of rasis 360 

131-142 Rising at other places 367 

143-151 Dasama lagna 372 
152-156 Rising of nirayana rasis in Orissa 374 

157-162 Charts and end 376 

C. TRIPRASNADHIKARA 

THREE PROBLEMS OF 380-475 

DAILY MOTION 

1-5 Cardinal directions 

6. Sanku and chaya 

7-10 Square root and multiplication 395 

11-13 Setting of sanku, correct chaya 402 

14-23 Definitions 

24-27 Kranti from palabha 

28-34 Sun from shadow and vice versa 410 

35-38 Unmandala sanku 412 

39-44 Agra, karna vrttagra 41 7 

45-51 Sama mandala 42j 

52-66 Kona sanku, natamsa 425 

67-80 Shadow form time & vice versa 430 



384 
394 



404 
408 



Subject Index v a 

81-84 Sun form agra & samasanku 437 

85 Path of shadow end 438 

86-87 Lapsed part of night 440 

88-92 Rising times in Orissa 441 

93-95 Conclusion and end 442 
7B. APPENDIX 

1. True, mean and standard time 443 

2. Equation of time 447 

3. Parallax 450 

4. Refraction 461 
8. LUNAR ECLIPSE 476-545 

TEXT 

1-6 Possibility of eclipse , 476 

7-11 Correct samaparva Kala 497 

12-15 Mean diameter etc. of sun, moon 500 

16-24 True diameter & distances 505 

25-27 Earth shadow 510 

28-33 Rahu and eclipse 513 

34-38 Moon sara, its gati 515 

39-40 Grasa, direction of eclipse 519 

41-43 Duration of eclipse 521 

44-45 Single step calculation 526 

46-54 Grasa at different times 528 

55-59 Direction of eclipse 531 

60-65 Lunar day length 536 

66-69 Valana correction 538 

70-77 Diagram 539 

78-83 True earth shadow 541 

84-88 Colour of eclipse 544 



Vlll 



9. 



Siddhanta Darpana 



10. 



11 



Solar Eclipse 

1-6 Lambana and nati 

7-15 - Sphuta lambana 

16-22 Single step method 

23-39 Nati correction in sara 

40-45 Tamomana, true moon bimba 

46-47 Hara 

48-49 Eclipse at different places 

50-57 True sthiti kala 

58-60 Single step method for duration 

61-62 Annular eclipse 

63-64 Reason for extra methods 

65-72 Eclipse duration through yasti 

73-82 Misc corrections 

83-85 Modern methods 

86-89 Duration limits and end 

PARILEKHA 
1-3 Valana 

4-5 Angular measure of bimba 
6-14 Diagram of eclipse 
15-30 Grasa calculation 
30-38 Another diagram, end 

CONJUNCTION OF PLANETS 
1-9 Meaning and methods 
10-11 Sara and pata 
12-26 Correction to sara 
27-31 Ayana drkkarma 
31-37 Aksa drkkarma 

» 

32-42 Diameter and bimba 
43-55 Mean and observed bimba 



546-599 

546 
547 
555 
558 
565 
570 
572 
573 
576 
576 
577 
577 
581 
583 
592 

600-612 
600 
602 
604 
607 
610 

613-651 
613 
618 
622 
624 
626 
630 
635 



Subject Index 




IX 


56-60 


Types of conjunction 


640 


61-67 


Nati 


641 


68-71 


Lambana 

* 


643 


72-90 


Conjunction with moon, stars 


643 


91-96 


Parilekha 


646 


97-108 


Observing planet shadows 


647 


109 


Increase of bright birhba 


649 


110-112 


Solar eclipse by sukra, end 


650 


12. 


CONJUNCTIONS WITH STARS 652-681 




TEXT 




1-11 


Longitude and latitude of stars 


652 


12-24 


Naksatra shape, no. of stars, yogatara654 


25-40 


Other important stars 


656 


41-56 


Saptarsi mandala 


659 


57-59 


Circumpolar stars 


666 


60-63 


Pole stars 


667 


64-75 


Arhsa of kala, mana, ksetra 


669 


76-79 


Sara of naksatra 


673 


80-87 


Conjunction & bheda 


674 


88-94 


Milky way, conclusion 


677 


13. 


RISING SETTING OF 682-707 




PLANETS, STARS 




1-6 


Daily and heliacal rising, setting 


682 


7-11 


Drkkarma for heliacal rising 


684 


12-25 


Kalarhsa of moon, stars, planets 


687 


26-29 


Rules for heliacal rising 


689 


30-33 


Days of rising • 


691 


34-37 


Aksa drkkarma of stars 

* 


692 


38-44 


Udaya dhruva of stars 


* 697 


45-50 


Sphuta Kalarhsa 


699 



x Siddhanta Uarpana 

50-58 Sphuta Udaya dhruva 700 

59-68 Kalaihsa of planets for Orissa 702 

70-82 Udayasta time & end 703 

14 - LUNAR HORNS 708-730 

TEXT 

1-13 Time of rising and setting of moon 708 

14-18 Moon at desired time (nata) 713 

19-27 Elevation of Horns 714 

28-29 Sara valana 715 

30-43 Diagram of horns . 719 

44-61 Modern method * 722 

62-63 Horns of budha, sukra 728 

64-69 Reasons of new methods, end 728 

15. Mahapata 731-754 

1-8 Two mahapata 731 

9-15 Calculation 734 

16-20 Mean value for current ayanamsa 736 

21-33 Pata from sphuta kranti 737 

34-42 Sparsa and moksa of pata 742 

43-58 Effects, duration 745 

59-62 Siddhanta methods 750 

63-66 Start of eras 751 

67-68 Importance of siddhanta etc. 752 

69-73 Dhruva, age of Brahma, end 753 

D. GOLADHIKARA 

!6. QUESTIONS ON METHODS 755-768 

TEXT 

1-4 Scope 755 

5-11 Importance of gola 757 



Subject Index 

16-20 
40-45 
46-49 
50-56 
57-65 
66-69 
70-78 
79-81 



17. 



18. 



1-12 
13-32 

33-38 

39-78 

79-93 

94-101 

102-104 

105-111 

112-115 

116128 

129-133 

134-142 

143-145 

147-161 

1-2 

3-22 

23-32 

33-38 

39-42 

43-44 



Doubts about earth motion 

Size of earth, orbit speeds 

Bhagana revision, guru years 

True planets, kranti, seasons 

Eclipse, conjunction 

Star light, size of brahmanda 

Meru, mahapata, horns, time units 767 

768 
769-811 

769 



XI 

759 
762 
763 
764 
765 
766 



Easy methods 
LOCATION OF EARTH 

Support of earth 

Earth as large sphere 

Bauddha view 

Motion of earth refuted 

Planetary motions from fixed earth 791 

Sara of planets 

Vapours of planets 

Centre of mass 

Effect of earth rotation 

Distance & revolution period 

Manda, sighra kendra 

Distance, motion of stars 

Darkness in night 

God's desire for fixed earth etc. 

DESCRIPTION OF EARTH 

Scope 



773 
777 
780 



^794 
795 
797 
799 
800 
802 
803 
805 
807 
812-887 
812 



Creation as per surya siddhanta 813 

Comments on creation 826 

Kala of moon, its water 827 

Light of stars, sun * 828 

Beings on planets 829 



847 
849 
850 



xu Siddhanta Darpana 

45-47 Composition of earth 829 

48-53 Jambu dvipa and meru 830 

54-81 Height & vision limit 839 
82-87 Sun set at meru 
88-91 Location of India 
92-98 Variation in oceans 

99-108 Meru 851 

109-152 Geography from Bhaskara 852 

153-167 Dimensions of earth, sphere 876 

168-175 Length units, end 885 

19. EARTH AND SKY 888-922 

TEXT 

1-6 Air spheres 888 

7-11 Nature of orbits 889 

12-21 Orbit lengths 890 

22-29 Linear motion of planets 892 

30-31 Sara gati 894 

32-37 Diagrams for true motion 895 

38-42 Less lambana of tara grahas 897 

43-52 Larger distance of sun 898 

53-58 Revision of orbits and naksatra Kaksa 901 

59-69 Measuring distance, diameter 902 

70-85 Height of hill, tree, cloud 906 

86-102 Vision limit for different heights 913 

103-108 Lords of days etc. 918 

109-125 Spread of light etc. 919 
20. INSTRUMENTS 923-971 

TEXT 

1-5 Scope 923 

6-44 Gola yantra of 2 and 1 axis 929 



Subject Index 




xm 


45-75 


Multiple axis yantra 


937 


76-77 


Kala yantra 


943 


78-81 


Golardha yantra 


944 


82-92 


Mana yantra 


948 


93 


Cakra, capa, turiya yantra 


957 


94-97 


Time measurements 


961 


98 


Phalaka yantra 


963 


99-108 


Svayamvaha yantra 


968 


110-113 


End 


971 


21. REMAINING EXPLANATIONS 


972-1025 


1-6 


Scope, day lengths 


972 


7-13 


Day night at poles 


973 


14-17 


Other day nights 


974 


18-22 


Rising time difference in rasis 


975 


23-26 


Ecliptic parts visible at places 


976 


27-31 


Unequal speeds, centre at sun 


976 


32-41 


Calculation from kalpa 


977 


42-54 


Bhagana corrections 


980 


55-70 


Correction after 10,000 years 


982 


71-78 


Guru years 


984 


79-83 


Padaka calculations 


985 


84-86 


Geocentric values 


986 


87-112 


Eccentric of moon 


987 


113-114 


Ucca kaksa 


992 


115 


Correction for moon 


993 


116-129 


Direct and retrograde motion 


993 


130-142 


Sighra and manda gati 


997 


143-146 


Revised methods 


1004 


147-151 


Bhaskara's manda, sighra gati 


1004 


152-163 


Ecliptic, nati, lambana 


1005 



22. 



23. 



xiv 

Siddhanta Darpana 

164-173 Vitribha „„*, 

17/1 ion t^ i 1007 

lln'iL A ^ dUrati ° n by dia ^ am 1008 

190-192 Ayana valana 1010 

lltlll P ° lar riSing ' m °° n horn ' niahapatalOll 
199-206 Importance of star circle ion 

207-224 Seasons, their zones 1012 

225-231 Formation of rain 1014 

232-248 Cube and cube root . lm * 

249-252 Conclusion ln I, 

1024 

E. KALADHIKARA 

SAMVATSARAS ETC 1026-1036 

TEXT 

1-15 Importance of kala 1026 

16-25 Time units from sun and moon 1028 

25-36 Nine measures, candra, naksatra 1029 
37-46 Solar times 

47-60 Sankrantis 1031 

61-70 Different years 1033 

71-77 Harivamsa units, river tide etc. 1034 

PURUSOTTAMA STAVA i 037 

(not translated) 

M- UPASAMHARA 1037 . 1078 

TEXT 

1-2 ^P 6 1037 

3-13 Kautuka panjika 1038 

14-29 Corrections to tithi, naksatra, yoga 1042 

30 Adhimasa 1047 

31-35 Naksatra, rasi of sun 1048 

36-38 Sun, moon speeds 1049 



Subject Index xv 

39-45 Possibility of eclipse 1050 

46-57 Tara graha in kautuka panjika 1052 

58-60 Pata and mandocca of moon 1058 

61-62 Complete revolution of planets 1060 

63-65 Soiar dates, use of Kautuka panjika 1061 

66-92 Topics in various chapters 1062 

93-139 Longitude, latitude of 109 places 1067 

140-141 Distance between places . 1073 

142-144 Purpose of book 1074 

145-149 Author and his family 1074 

150-160 Conclusion 1076 
24A Appendix (Sanskrta terms) 1079-1094 
Name & subject index 1096-1115 



INTRODUCTION 

(1) Arrangement of the book - 

Scope - Original book was written in 2,500 
Sanskrit verses in Oriya script on palm leaves. It 
was published with introduction in English by Prof. 
Jogesh Chandra Roy of Ravenshaw College, Cuttack 
by Calcutta University in 1899. Subsequently same 
edition was reproduced with approximate Oriya 
translation by Pandit Vir Hanumana Shastri, by 
Utkal University, Bhubaneswar (then at Cuttack). 
This was reprinted by Dharmagrantha Stores, 
Cuttack. Some parts have not been translated and 
explained. First volume of this book renders the 
Sanskrit verses in devanagari script with literal 
Hindi translation. It also contains the original 
introduction. 

. Present second volume translates all the verses 
in English. Translation is not literal but in 
mathematical terms, but preserving the technical 
terms in Sanskrit. Verses in praise of god have 
been left out, not because of disrespect. With all 
devotion inspired by Samanta Chandrashekhar, this 
is not the purpose of the second volume. In addition 
to translation, each formula has been explained or 
derived according to modern mathematics and 
astronomy. The methods have been compared with 
other Indian astronomers and some times with 
other countries and with modern astroftomy. This 
was the method and purpose of Samanta himself. 



2 Siddhanta Darpana 

Technical terms and their calculations cannot 
be explained in words alone. So a general 
mathematical and technical introduction is given at 
beginning of each chapter with bibliography or 
source reference for further study. In that light 
only, the methods proved in the chapter can be 
understood. Where-ever considered useful, 
methods have also been explained with examples, 
based on text as well as modern astronomy. 

In Sanskrit verse, some number or statement 
has been continued in many verses due to poetic 
and literal explainations. They have been clubbed 
together for translation. For brevity and simplicity, 
many parts have been given in chart form. Chapter 
23 contains only verses in praise of god. Most of 
these verses have two or more meanings. It cannot 
be expressed in other language, nor it is related 
to the main topic. It is, therefore, omitted.* 

2. Numeration 

Decimal system of writing numbers originated 
in India. Arabs called them Hindu numerals. 
Europeans learnt from Arabs and termed them 
Arabic numerals. This system uses 10 symbols 0, 
1, 2, 3, 4, 5, 6, 7, 8 and 9, each increasing by 
one. For writing greater numbers, successive 
positions towards left areused, each place having 
ten times the value of position on its right side. 
Similarly, fractions are written towards right from 
the unit place after giving a point, called decimal. 
Each place has value of l/10th of the value of its 
predecessor towards left. 

Modern computers use binary system with 
two symbols and 1 only, each place value 



Introduction 3 

increasing two times towards left. In angular and 
time measurements of Indian astronomy, continued 
till today, multiples or divisions by 60 at each step 
is used. This was used in Sumerian mathematics 
for all numbers and is called sexa-gesimal (60) 
system. 

Aryabhata, I, has given the following order 
of place values, each ten times the preceding - 

Eka (units place), Dasa (ten place), Sata 
(hundred), Sahasra (thousand), Ayuta (ten 
thousand) Niyuta (hundred thousand or lakh), 
Prayuta (ten lakhs or a million), Koti (ten millions 
or 1 crore) Arbuda (10 crores), Vrnda (100 crores) 
etc. 

Sankara Varman in his Sadratnarhala (1,5-6) 
has given the following sequence in multiples of 
10 - 

Eka (1), Dasa (10), Sata (100), Sahasra (1,000), 
Ayuta (10,000), Niyuta (or lakh, 10 5 ), Prayuta (10 6 ), 
Koti (10 7 ), Arbuda (10 8 ), Vrnda (10 9 ), Kharva (10 10 ), 
Nikharva (10 11 ), Mahapadma (10 12 ), Sanku (10 13 ), 
Varidhi (10 14 ), Antya (10 16 ) and Parardha (10 17 ) 

Lalita-vistara, a Buddhist text gives powers of 
10 beyond 100 koti (i.e. 10 9 ), each increasing 100 
times the previous - 

Koti (10 7 ), ayuta (10 9 ), niyuta (10 n ), kankara 
(10 13 ), vivara (10 lS ), aksobhya (10 17 ), vivaha (10 19 ), 
utsanga (10 21 ), bahula (10 23 ), nagabala (10 25 ), 
titilambha (10 27 ), vyavasthanaprajnapti (10 29 ), 
hetuhila (10 31 ), karahu (10 33 ), hetvindriya (10 35 ), 
samaptalambha (10 37 ), gananagati (10 39 ), niravadya 
(10 41 ), mudrabala (10 43 ), sarvabala (10 45 ), visajnagati 
(10 47 ), Sarvasajna (10 49 ), vibhutangama (K? 1 ), 
taUaksana (10 53 ). 



4 Siddhanta Darpana 

Aryabhata notation - Varga letters (k to m) 
should be written in varga places (unit place and 
hundred times at each step) and avarga letters (y 
to h) in the avarga places. Varga letters take the 

numerical values (1,2,3 25) from k onwards. 

Numerical value of the initial avarga letter y is n 
plus m (i.e. 5+25), next letters are 40 to 90. In 
nine places of double zeros, nine vowels should 
be written (one vowel for each pair of varga and 
avarga letters). 

Katapayadi notation was before Aryabhata and 
is believed to have been used in vedas in portions 
related to astronomy or mathematics. It was very 
popular in Kerala. Each digit is represented by a 
consonant letter. Vowels and half letters have no 
meaning. Digits are written from right to left to 
form a number. Numbers 1 to nine and the are 
indicated by letters starting from k, t, p, or y, 
hence the system is called katapayadi. 



1 


2 


3 


4 


5 


6 


7 


8 


9 





k 


kh 


g 


gh 


n 


c 


ch 


ja 


jh 


n 


t 


th 

• 


d 

• 


dh 

• 


n 

• 


t 


th 


d 


dh 


n 


P 


P h 


b 


bh 


m 












y 


r 


1 


V 


s 


s 

■ 


s 


h 


1 





Suryasiddhanta and other works including the 
present book have used words to indicate each 
digit again written from right to left. These have 
already been indicated in the 1st volume for 
purpose of literal hindi translation/ and need not 
be repeated here. 



Introduction 



3. Transliteration of Sanskrit letters 






vowels 












Short 


3T 


? 


3 


3s 


"& 






a 


1 


u 


r 

• 


1 




Long 


3rfT 


i 


3i 


^ 


atr ^ 


4 




a 


1 


6 


e 


o ai 


au 


Anusvara 


• 


m, m 








Visarga 


* 


h 

• 












Consonants 












^ 


n. 


T 


\ 


^ 






k 


kh 


g 


•gh 


n 






S 




^r 


H 


^ 






c 


ch 


* 

J 


jh 


n 






z 


3 

^ 




• 


TTf 


* 




t 

ft 


th 

• 


d 

* 


dh 

* 


n 

• 






*t 


1 


\ 


H 








t 


th 


d 


dh 


n 




* 


\ 


T 


*t 


1 


1 






p 


ph 


b 


bh 


m 




\ 




^ 


^ 


*t 


1 ■ '* 


* * 


y 


r 


1 


V 


s 


m s 

• 


h 1 



4. A brief survey of Indian astronomy 

Astronomy has come from old French word 
'astronomie' which in turn was derived from Latin 
'astronomia' and Greek 'astronomos' - meaning star 

law. 

'Jyotisa' in Sanskrit means the same - 'Jyoti' 
means source of light i.e.a star in a sky; study of 
star groups and motion of planets observed 
through them is jyotisa. 



6 Siddhanta Darpana 

Greek astronomy had its origin in Nile river 
and Sumerian civilisation. Western astronomers try 
to establish that vedic jyotis is originated from 
Sumer and later Indian astronomy is influenced by 
Greek. But internal astronomical evidence suggests 
that text of vedanga jyotisa was written in 2976 
B.C. when summer solstice started (verse 6 tells 
that Magna month began when Sun was in mid 
Aslesa. Full moon point was 1°13' east of mid 
magna i.e. 8° east of Regulas at present at 150° 56' 
- difference of 68° 56'). However, system is much 
older, and many changes have been made from 
Taittiriya Samhita. 

Vedanga jyotisa is found in two texts - Rkveda 
has 36 verses on the topic and yajurveda has 43 
verses. Many are common, but the system is 
entirely different. Yajur jyotisa was written 624 
years after Rk jyotis according to internal evidence. 
Compiler of Rk jyotis is 'Lagagha.' According to 
difference in day lengths, mentioned in verse 7 
and 22, they refer to a place of 34° 50' North 
latitude. In northern borders of India, this is near 
Alma - Ata of Kyrgiz. Since it was first seat of 
learning, first school is called alma-meter is Greek. 

So far, authors have assumed that both 
versions of vedanga jyotisa denote 5 years yuga 
(or cyclic period). Accordingly text of Rk jyotisa 
had to be modified and twisted. But now Sri P.V. 
Holay of Nagpur in his Vedic Astronomy (1988) 
has proved that the original text of Rk jyotisa 
indicates a 19 year yuga - after which solar and 
lunar years start together. There are 7 extra months 
in a yuga, their adjustment is such that 5 solar 
years start within 6 days of new moon. Such 



Introduction / 

approximately concurrent years are called 
Samvatsara. Other types of years are Anuvatsara, 
Parivatsara, Idvatsara and Idavatsara. Thus the 
statement that a yuga has 5 samvatsaras doesn't 
mean that 5 years make a yuga as assumed so far. 
It means only that out of 19 years in a yuga, 5 
are samvatsaras. 19 years cycle was later on 
discovered by a Greek astronomer Metori in 432 
B.C. and is called Metonic cycle. However, this 
cycle was used by Sumerians and Chinese also in 
their calender much before the Greeks. It is certain 
that astronomy in the whole world had single 
system. Irrespective of origins, there was exchange 
and compilation of ideas, and same standard was 
adopted as in the modern sciences. Thus houses 
of zodiac and constellations have the same names 
in all the languages. There is similar correspon- 
dance in medical names of Greek origin and their 
Sanskrit names in yoga or ayurveda. 

Vedanga Jyotisa was followed by Garga 
Samhita and Paitamaha Siddhanta and Jain works 
Surya-pannati and Jyotiskarandaka with minor 
changes. This period was followed by so called 
Siddhanta period. According to traditional Indian 
belief, there were 18 such siddhantas - (1) Surya 
(2) Paitamaha (3) Vyasa (4) Vasistha (5) Atri (6) 
Parasara (7) Kasyapa (8) Narada (9) Gargya (10) 
Marlci (11) Manu (12) Angira (13) Lomasa (or 
Romaka) (14) Paulisa (15) Cyavana (16) Yavana (17) 
Bhrgu and (18) Saunaka, Five of these siddhantas 

- Saura, Paitamaha, Vasistha, Romaka and Paulisa 

- were codified by Varahamihira in his 
Pancasiddhantika (184 B.C.) who has^ emphasised 
that the Saura was most accurate of them. 



8 Siddhanta Darpana 

Saura or Surya Siddhanta has no human 
authorship. Second verse of the text states that 
when short time (or 121 years in Katapayadi) was 
remaining in end of Satyuga, Sun god taught this 
to Maya asura. Yuga system of this originates from 
Visnudharmottara purana according to Brahmagup- 
ta which is modification of old Brahma (or 
Paitamaha) siddhanta. 

In Varahamihira's Saura, a period of 180,000 
years has been stated which contains 66,389 inter 
calary months and 10,45,095 ommitted lunar days 
(tithis). Modern Surya Siddhanta tells about a 
mahayuga (or yuga) of 43,20,000 years divided into 
Krta, Treta, Dvapara and Kali ages in ratio of 4:3:2:1 
(12,000 divine years) with 1/12* period each in 
beginning and end as sandhya (twilight period). 
360 solar years are called a divine year. Paitamaha 
siddhanta is crudest and has 5 years yuga like 
yajus jyotisa. Vasistha has improvement and deals 
with true motion of 5 planets. Sidereal year has 
been stated of 365 1/4 days. 

Paulisa siddhanta is more accurate and gives 
days counts (ahargana) and sine tables. It gives 
solar year of 365.2583 days. Al-Barurti has regarded 
Paulisa as a Greek from Alexandria (Sachau I, 
p/153). 

Romaka gives a luni-solar cycle of 2850 years 
with 1,050 intercalary months and 16,547 omitted 
lunar days. Length of year is 365 days 5h 55' 12" 
and synodic month is 29 days 12h 44'2.2". It deals 
with equations of centre for Sun and mooni 

Among present compiled texts, Aryabhatiya 
of Aryabhata I (476 A.D.) of 121 verses is the first. 



Introduction 9 

* 

It is a brief codification of existing knowledge after 
observatory (khagol village) near Kusumpur 
(modern Patna, capital of Bihar) was destroyed in 
Huna attack. It is more an attempt to preserve the 
science in verse form, than to write a text book. 
For brevity, he has devised his own number 
system, as explained before. Subsequent 
astronomers made appropriate corrections and 
devised simpler methods of calculations in their 
texts. 

Jyotisa has three parts - (1) Ganita - cor- 
responding to modern astronomy and mathematical 
methods (2) Phalita - Astrology (3) Hora or Samhita 
- auspicious times, natural phenomena, signs in 
human beings and animals etc. Ganita' is written 
in three styles - (1) Siddhanta is a text for 
calculation from beginning of yuga. (2) Tantra starts 
the calculation from beginning of Kaliyuga (17/18- 
2-3102 B.C. Ujjain mid-night) (3) Karana uses short 
methods for current years ephemeris with reference 
to a recent base year. Its literal meaning and use 
is same as that of a handbook or a manual. 

A brief list of astronomers and their works is 
indicated below - 

5th-6th Century - Aryabhata I (Aryabhatiya 
and Aryabhatasiddhanta, the later available only 
in quotations). 

6th Century - Prabhakara, pupil of Aryabhata, 
Varaha-mihira (Pancasiddhantika and Brhatsa- 
mhita). 

6-7th Century - Bhaskara I (Mahabhaskariya, 
Laghubhaskariya and Aryabhatiya - bhasya); Brah- 
magupta (Brahma-sphuta-siddhanta and Khanda- 



10 Siddhanta Darpana 

khadyaka), Haridatta (Grahcara-nibandhana) 
Devacarya (Karana-ratna). t 

8~9th Century - Lalla (Sisya-dhi-vrddhida- 
Tantra) Govinda - Svamin (Mahabhaskariya-bhasya) 
Sankaranarayana (Laghu-bhaskariya vivarana) 
Prthiidaka svamin (Brahma-siddhanta vasana 
bhasya) and Khanda-Khadyaka vivarana. 

10th Century - Vatesvara (Vatesvara- 
siddhanta) Munjala (Laghumanasa), Sripati 
(Siddhanta-sekhara) Aryabhata II (Mahasiddhanta), 
Bhattotpala (Khanda-Khadyaka vyakhya and 
Vrhatsamhita - vyakhya) Vijayanandin (Karana 
Tilaka). 

11th Century - Somesvara (Aryabhatiya 
Vyakhya) Satananda (Bhasvati) 

12th Century - Bhaskara II (Siddhanta 
Siromani with Vasana bhasya, Karana Kutiihala), 
Mallikarjuna Suri (Surya siddhanta Vyakhya) 
Suryadevayajvan (Aryabhatiya Prakasika and 
Laghumanasa Vyakhya) Candesvara (Surya- 
Siddhanta Bhasya). 

13th Century - Amraja (Khanda-Khadyaka- 
Vasana bhasya) 

14th Century - Makkibhatta (Ganita Bhusana), 
Madhava of Sangamagrama (Sphuta candrapti, 
Aganitagrahacara, Venvaroha), Madanapala 
(Vasanarnava on the Surya siddhanta), Viddana 
(Varsika Tantra), 

15th Century - Paramesvara (Drgganita, 
Goladipika) Grahanamandana, Grahana-nyaya- 
dipika, Aryabhatiya vyakhya, Bhatdipika, 
Mahabhaskariya vyakhya, Laghubhaskarlya 
vyakhya, Surya siddhanta-vyakhya and 



Introduction ** 

Mahabhaskariya bhasya vyakhya) Yallaya 
(Aryabhatlya vyakhya, Jyotisa darpana, 
Laghumanasa-kalpataru arid KalpavalK on the 
Surya-siddhanta), Rama Krsna Aradhya (Surya 
siddhanta Subodhini) Cakradhara (Yantra 
Cintamani) Nilakartha Somayaji (Jyotirmlmansa, 
Golasara, Candracchayaganita, Siddhanta Darpana, 
Tantra Samgraha and Aryabhatlya bhasya). 

16th Century - Jyesthadeva (Yuktibhasya, 
Drkkarana) Sankara Variyar (Karana sara, Tantra- 
sangraha, Yukti dipika), Bhudhara (Surya 
siddhanta vivarana) Tamma yajvan 

(Grahanadhikara, Surya siddhanta - Kamadogdhri), 
Ganesa Daivajna (Grahalaghava, Tithi-Cintamani, 
Pratodayantra and Siddhanta Siromani-Vyakhya), 
Acyuta Pisarati (Karanottama, Sphutanirnaya with 
vivarana, Uparagakriya - Krama, Rasigola 
sphutaniti); Rama (Rama vinoda) 

17th Century - Visvanatha (Grahanartha 
Prakasika, Grahalaghavatika, Karanakutuhala 
Udaharana) Candidasa (Karana Kutuhala TTka), 
Putumana Somayaji (Karana paddhati, Pancabodha, 
Nyayaratna) Nityananda (Siddhantaraja and 
Siddhanta sindhu) 

18th Century - Maharaja Sawai Jayasimha 
(Yantraraja racana, Jayavinodasarani), Jagannatha 
Samrata (Samrata Siddhanta) 

19th Century - Sankaravarmana 
(Sadratnamala) 

For easy calculation of pancanga, many 
astronomical tables have been prepared. These are 
called Kosthaka or Sarani. Early examples are 
Grahajnana by Asadhara^ (epocb-20-3-1132), 
Laghukhecara siddhi by Sridhara (20-3-1316), 



22 Siddhanta Darpana 

Makaranda by Makaranda (epoch 27^3-1478) Kheta- 
muktavali by Nrsimha (31-3-1566) 

(5) Astronomers of Orissa 

Orissa was part of the Indian tradition of 
Jyotisa from vaidic and siddhanta period. 
Astronomy and mathematics were related to Yajnas 
whose time was found with astronomy and 
construction was as per geometric diagrams. In 
Orissa, Brahmanic titles related to yajnas still exist 
like - Hota; Udgata, Brahma, Pathi, Pari, Vagmi 
etc. It is quite probable that Taittiriya Samhita and 
Aranyaka, Aitareya and Gopatha Brahmana etc. - 
the aranyaka granthas forming origin of astronomy 
flourished in places like western Orissa which were 
famous as aranya or mahakantara. Another indica- 
tion of rise of astronomy is the sea trade from 
Orissa coast to East Asia and upto Roman Empire. 
Due to popularity of Bali yatra, it is thought that 
sea trade of Orissa was only with Bali - a small 
island in Indonesia. However, the relations must 
have developed with other areas of South East 
Asia and Chinese coast and intermediate islands 
of Andamana group must have formed base for 
supply of food etc. Late Dr. N.K. Sahu in his 
history of Orissa states that silk of Sambalpur was 
known in Roman empire also. This confirms that 
ships from Orissa and other parts of India were 
going to different parts of the world. Technically, 
visit to America was also possible and traditional 
jyotisa texts mention a town 90° east of Ujjain 
(yamakoti) which should be in New Zealand 
(southern hemisphere). Hence yama is lord of south 
direction. 180° east of Ujjain in Siddhapura in 
North hemisphere. At this longitude there is a 



Introduction x ^ 

town near Mexico where greatest Pyramid was built 
- Valmiki Ramayana calls it a gate built by Brahma 
to indicate end of east direction i.e. 180° East of 
Ujjain at prime meridian in Indian Astronomy 
(Kishkindha kanda). To a layman this discussion 
appears irrelevant to astronomy. However, sea 
journey (and plane journey in modern times) is 
not possible without knowledge of astronomy. 
There are no landmarks in sea or sky for finding 
the way. Hence navigation requires accurate deter- 
mination of longitude, latitude and direction. These 
three are discussed in an important chapter 
Triprasnadhikara' of Surya siddhanta. It is note- 
worthy that Columbus could undertake his journey 
in open sea only because method of finding 
longitude was discovered in western astronomy ten 
years before that. That was from Turkish ships 
who had learnt astronomy from India. Vice versa, 
longitude determination in remote past indicates 
that India was well versed in navigation round the 
globe. 

Transport of rice from Orissa was marked by 
Salivahana Saka in 78 A.D. - it means transport of 
rice (Sali = rice, Vahana - carriage). As a product 
of Audra (Orissa), rice was called Audriya i.e. 
Oryza in greak. This has become rice in English 
(omitting '0') and Orissa as name of the state. 
Navigation history indicates traditional study of 
astronomy in Orissa. 

Surya Siddhanta has been given by sun god, 
whose worship is most common in coastal areas 
and river ports in India and elsewhere (Japan, 
Egypt, Mexico, Peru etc.). Jyotisa study might have 
suffered during Buddhist era in Orfesa. It again 



14 Siddhanta Darpana 

picked up after Varahamihira in orissa like other 
parts of India. Gariga period (650 to 900 A.D.) 
records of Orissa indicate that Brahmans were well 
versed in Vedanga of which jyotiSa is a part. One 
person has specifically been mentioned as 
siddhanti. Satananda was most famous of old 
astronomers of Orissa. 

Satananda - He was son-of Sankara and 
Sarasvati of Purushottampura (Puri) who completed 
his famous work Bhasvati"' in 4200 yugabda (1099 
A.D.). He has made calculations with reference to 
Purl. Full name of Bhasvati was Panca siddhanta 
sara or Pancasiddhanta - Bhasvati on pattern of 
Pancasiddhantika of Varahamihira. However, he 
has followed Surya siddhanta only which is 
considered most accurate. It is a Karana grantha 
following solar year starting from Sayana mesa 
samkranti. It was popular for its accurate calculation 
of eclipse - UlpJt*ii*qal«RJT Commentaries on Bhasvati- 

(1) Sansaraprakasika by Kasisekhara 

(2) Balabodhini TIka of Bhasvati in 1543 A.D. 
by Balabhadra son of Vasanta of Kausika gotra in 
Uma town of Jumila state. 

(3) Oriya translation by Trilocana Mohanti in 
Yugabda 4747. Other books of 'Satananda are - (1) 
Satananda Ratnamala - a samhita book like 
Ratnamala of "Sripati, his elder contemporary. 
(Palm leaf manuscript No 268, Orissa Museum). 
(2) Satananda Samgraha - work on smrti. No 
manuscript is available. 

Only Bhasvati is available with Hindi com- 
mentary by Matr Prasada Pandey by 
Chaukhambha, Varanasl. 



Introduction 1$ 

Other astronomers of Orissa - 

(1) Jayadhara Sarma of Kotarahanga near 
Sakhigopala (Purl) received grants from Bhanja 
kings in 1231-12;* 5 A.D. for his mastery on Jyotisa. 
Though he was famous, no book by him or his 
forefathers is available. 

(2) Gajapati Kapilesvara Deva (1435-1466 A.D.) 
of Cuttack who started Kapila era got another book 
written after his name called Kapila Bhasvati. But 
no manuscript is available. 

(3) Govinda Dasa of Nagesa gotra son of Hira 
Devi was a great astrologer. He constructed a 
dola-mardapa in sacred town of "Sri Kurma". No 
work by him is available. 

(4) Trilocana Mahanti - He translated Bhasvati 
in Oriya verses in 4747 yugabda (1646 A D). 

(5) Gajapati Narayana Deva of Parla Khemun- 
di wrote Ayurdaya Kaumudi in 26 chapters around 
1650 A.D. 

(6) Vipra Namadeva - He wrote a samskrta 
commentary Sarvabodhini on Suryasiddhanta in 
1721 AD. 

(7) Dhananjaya Acarya - wrote a Palaka 
Panjika for 1665 Saka (1733 A.D.). 18 chapters of 
his Jyotisa candrodaya are available in Orissa 
museum. He wrote another work Jataka 
Candrodaya. 

(8) MagunI Pathi, son of Markandeya Pathi 
wrote an Oriya commentary Mandartha bodhini 
on Siddhanta Siromani in 1741 A.D. In 1744 A.D. 
he wrote another commentary in Oriya on 
Grahacakra of Kocanacarya. m 



16 Siddhanta Darpana 

There is an incomplete work Jyotis Sastra by 
Markanda who may be his father. 

(9) Mahamahopadhyaya Dayanidhi Nanda 
wrote Sisubodhini in 1707. 

(10) Mahamahopadhyaya Chapadi Nanda 
wrote Balabodharatna Kaumudi in 1763. 

(11) Son of Srinivasa Misra wrote Jyotis tattva 
Kaumudi in 18th century. First 12 chapters are 
available. 

(12) Gadadhara Pattanaik S/o Padmanabha in 
18th century wrote Ravindu grahanam on basis of 
Kocana-carya in 18th century. 

(13) Gopinatha Dasa (Patnaik) wrote Ayurdaya 
Siromani and Suddhahnika Paddhati. 

(14) Caitanya Raja Guru - wrote 
Laghusiddhanta on pattern of Surya Siddhanta and 
wrote one Oriya commentary on it. 

(15) Yajna Mishra S/o Visyambhara wrote 
Jyotisa Cintamani or Ratnapancaka whose incom- 
plete manuscripts are available. 

(16) Mahidhara Mishra wrote Mahldhara 
Samhita in 18th century and a commentary on 
Amarakosa. 

(17) Prajapati Dasa - (Unknown time) - 
Grantha Samgraha, pancasvara and Saptanga. 

(18) Bhanusekhara Dasa (18th century) Tarani 
Prakasika , a commentary on Jataka Ratnakara. 

(19) Dasarathi Mishra (18th century) - Jyotisa 
Samgraha. 

(20) Krsna Misra (18th century) - Naksatra 
Cudamani, Kala Sarvasva. 



Introduction 17 

(21) Tripurari Dasa - Oriya poet of 17th 
century - He wrote the following books on Kerala 
astronomy - Kerala Sutra, Kerallya dasa and Prakrta 
Kerala. 

(22) Nilakantha Praharaja and his son Yogi 
Praharaj - Their books Smrti Darpana and 
Vaidyahrdayananda have been published by 
Madras Govt. 

(6) Samanta Candrasekhara and his role 

Brief Biodata - He was born on 11.1.1936 
(Tuesday) i.e. Pausa Krsna 7/8th 1892 Vikramabda 
(1757 saka) For an astronomer it is proper to give 
his birth time by planetary positions which is free 
of a calendar system. 

Birth time - 09-04 1ST based on Kumbha lagna 
and dasa calculation 

Birth place - Khandapara (Purl ) 

Latitude 20° 15' North, longitude 85°6'East 
Lagna 310° 40' (Pranapada in 5th house, Navamsa 
is Makara. 

Ayanamsa 21°34' 

Sun 268° 28' Moon 172 Q 35' 

Mars 263° 26' Mercury 271° 45' 

Jupiter 77 '40' Venus 292° 25' 

Saturn 192° 46' Rahu 34° 56' 

Uranus 306° 59' Neptune 281° 31' 

Balance of Moon dasa of birth - 6 months 23 
days. 

Important events of his life - He did not 
have formal university education. Even though he 
was born in a royal family, he sufferred poverty 



^ Siddhanta Darpana 

and unhappy family throughout his life. He 
sufferred from chromic dyspepsia and stomach 
inflammations frequently. At the age of 22, he 
married princes Sita Devi of Angula Raj family. 
Due to his ugly looks his father-in-law showed 
reluctance to give his daughter in marriage in lagna 
mandapa. When he showed his deep knowledge 
of Sastras and mastery over Samskrta verses, his 
marriage was solemnised (possibly on 28-2-1858). 
He had 5 sons and 6 daughters, out of which two 
sons had expired. He was banished from 
Khandapada by his ruler, being his own cousin. 
But due to his knowledge, he gained wide fame 
and his rights were restored by the then commis- 
sioner of Cuttack. He was also given a 'Sanad' in 
honour of his achievements. His work "Siddhanta 
Darpana" cannot be fully understood by a person 
unless he is well versed in Indian astronomy as 
well as modern mathematics. Whatever is known 
to common public about the book or its author is 
based on the English introduction by Prof. Jogesha 
Chandra Roy. This is based on personal interviews 
and not on a stfudy of the book. So, many vital 
points have been left out. Samanta expired on 
11.6.1904. On the basis of -his horoscope he had 
foreseen his death; which is expressed by his son 
Gad&dhara in an Oriya verse-meaning - "father 
called me near and told that moon had entered 
his maraka naksatra, and there was no escape from 
death". In last but one verse of Siddhanta Darpana, 
he has expressed desire that his body should fall 
at the feet of Lord JagannStha. On his last day, 
he went for darshana of lord. At the time of bowing 
before Jagannatha, he expired. At every place in 



Introduction ^ 

the book, he has shown his deep faith in lord and 
the scriptures. He has accepted his experimental 
observations only when they found support in 
some scripture. 

His works - Siddhanta Darpana is work of 
his whole life. At the end of every chapter two 
fold purpose of this book is explained - (1) 
Balabodha - i.e. a text book and (2) Ganita-Aksi 
Siddhi - i.e. tally of calculation and observation. 

For text book purpose, this is a treatise on 
Indian astronomy containing relevant positions of 
all text books from Sakalya Samhita to siddhanta 
books starting from Aryabhata. Quotation from 
Atharvaveda is unique in Indian astronomy; as it 
is only correct figure for sun's diamater,in Indian 
astronomy or in western astronomy before advent 
of telescope about 300 years ago. It is most 
voluminous book on astronomy with 2500 verses. 
Next largest are Vateswara Siddhanta with 1100 
verses or Siddhanta Siromani with 900 verses. 

Ganita-Ak§i Siddhi has three fold significance. 
As every other science, purpose of astronomy is 
to tally mathematical calculations with observations. 
Books starting with Aryabhata have only 
formulated or coded the existing knowledge, they 
have not indicated source of such figures. Methods 
are often in-completely explained and only refer to 
Vedic origin which is not clear. Thus purpose of 
math is only to find calculation methods for finding 
the observed position. Siddhanta books are not 
concerned about mathematical models, theories of 
gravitation or theories of motion. We are satisfied 



20 Siddhanta Darpana 

when calculations give correct result and not 
bothered whether Sun or earth is centre of motion. 

Third aspect is that there is slight change in 
planetary motion over long periods of time as 
stated in Surya siddhanta. This happens due to 
tidal friction. But siddhanta texts after Aryabhata 
have assumed constant motion throughout yuga or 
a kalpa of 1000 yugas. Due to approximation of 
constants or errors in calculation methods there is 
some deviation in observed results. In every period 
astronomers have corrected the constants given in 
Surya Siddhanta. according to need. These are 
called Bija corrections. 

Researches of Candrashekhara : 

(1) Moon's Motion (a) Traditionally moon's 
equation was of the form - 

300'49.5," Sin (nt-a) + 2'23.25" Sin $. (nt-a) 

2nd term is equation of apsis introduced by 
Brahmagupta. This form is correct, but constant is 
slightly wrong. 

(b) Sripati had found effect of Sun's attraction 
on moon motion (called evection). This has been 
introduced as Tungantara correction by 
Candrasekhara given as 

-160' cos(0 -a) Sin (D - 6) X 

Moon's apparent daily motion 
daily mean motion 

Error is about 4' only. 

(c) Bhaskara II had observed a fortnightly 
variation in moon's motion giving an error of 
maximum of 6 dandas in middle of pak§a. 



Introduction 21 

Comparing with his own observation, Samanta 
gave the Paksika equation as 

38'12" Sin 2 (D-d) 

where D' is moon corrected by 1st and 2nd 
equation 

(d) Due to Sun's annual motion, a digamsa 
correction also was introduced. 

±11'27.6" Sin (Sun's distance from apogee) 

These equations almost give the modern value 
and are to be further checked after 1000 years. 

(2) Ayanamsa - According to modern theory, 
earth's axis is rotating in a conical motion complet- 
ing almost uniform circular motion in 25,726 years. 
Samanta has assumed libration theory that the 
motion is not circular with 360" rotation but 
pendulum like oscillations within values of 27 ° , 
but with uniform motion. Present value of 
Ayanamsa tallies with both theories. Only after 300 
years or so error may be noticed. He has corrected 
the value of Surya siddhanta slightly (6,40, 170 
revolutions in a kalpa, instead of 6 lakh revolutions 
according to Surya siddhanta). Liberation theory is 
not supported by modern astronomy but it may 
be correct according to methods of projective 
geometry used in Jain Astronomy (Thesis by Sri 
S.S. Lishk). 

(3) Mandocca gati - According to classical 
mechanics, planets move in elliptical orbit whose 
major axis is fixed in space. Partly due to acti 
of other planets (mainly jupiter) and partly due to 
general theory of Relativity (1917 - Einstein), force 
of attraction reaches at speed of fight, not 



22 Siddhanta Darpana 

instantaneously - mandocca (apogee) is moving 
slowly. For mercury, it was calculated to be I* in 
11,000 years which was tested in 1919. In 300 years 
since Tycho Brahe, it is only 1/36° of an angle. 
For other planets it is so slow that it cannot be 
measured even by modern instruments. Indian 
astronomy gives V in 12,000 years for mercury 
and 39 revolutions of Saturn in 1 Kalpa or 1° in 
3 lakh years. 'Cosmology' by Narlikara gives its 
rate of movement as 





6 Jt 


G M 




L 


Tc 2 



where M = mass of Sun, T = period of planet 
Samanta has introduced a new concept of Parocca 
for Mars and Saturn which moves with constant 
circular motion around which mandocca oscillates. 
This is not supported by relativistic equation. But 
it may be probable due to effect of Jupiter between 
Mars and Saturn, which can be tested only by a 
computer calculation. Another doubt is that such 
a motion cannot be observed in one life time. Even 
Moon's equation of motion is based on 1000 years 
of observation and needs same time more to test 
it. Samanta has not mentioned the basis of his 
correction. 

(4) Discussion of other theories - Prof. J.C. 
Ray had not read Siddhanta Darpana and wrote 
introduction on basis of personal discussions. But 
Samanta has treated him as student and has 
criticised his opinious about modern physics in his 
chapters on discussions (Vasana-rahasya). 



23 
Introduction 

(a) Jain theories have been criticised because 
they were based on projective geometry and 
become absurd according to spherical geometry. As 
a single* sphere of earth is drawn as two circular 
maps in projective geometry, two Suns and two 
Moons were assumed in Jain theories. But dimen- 
sions of imaginary mountain 'Meru' have been 
quoted on the basis of Jain theories only. 

(b) In Indian astronomy, for calculation pur- 
pose, it is immaterial whether earth in fixed or it 
moves. In both views, relative motion will be same 
riving the same result. Samanta has used modern 
physics to refute the theory that Sun is not the 
centre of motion. It is mass centre of solar system, 
which is away from Sun's surface at a distance of 
l-l/2times its radius in direction of Jupiter (effect 
of other planets can be neglected). It appears that, 
Samanta was too skeptic of European theories 
whom he has called 'golden theory' as they were 
supported with hope of getting gold medal (17-160) 

(c) His other objection was that if earth moves 
on its axis; why Jupiter moves faster being the 
heaviest. This has been explained later on by 
presuming that Jupiter and Sun were twin stars. 
Due to loss of matter, Jupiter gained m angular 
speed, to preserve the momentum. The other 
objection as to why we observe the same side of 
moon - has not been explained so far. 

(d) If stars are all like Sun and are equally 
spaced in all directions, there should be no day 
and night - every time equal light should come 
from all directions. This was called Olber's paradox 
in modern astronomy and was explained only m 
1930 when expansion of universe was observed (It 
is mentioned in Indian scriptures also) Due to 
expansion, the farther stars have lesser effect and 



24 Siddhanta Darpana 

only the Sun causes day and night. Samanta has 
correctly refuted the argument of absorption of 
light of stars by gases etc. 

(5) Diameter of Sun : Diameter of Sun had 
been heavily under estimated to be about 10 to 14 
times the diameter of moon by all astronomers in 
India and outside. After talescope it was known 
to be 400 times. Distance of Sun being 400 times 
that of Moon, it will cause much greater difference 
in amount of solar eclipse at two places. This might 
have prompted Samanta to correct it. But he has 
referred to Brahmavidya upanisad and Atharva 
veda (describing expansion of 3*) to get the value 
of 72,000 yojans (19-40,50) and (8-12). This become 
162 times diameter of Moon. 

Siddhanta Darpana has taken value of 1 yojans 
as 4.9 miles. Had. he taken it to be 11 miles 
(Aryabhata 7.5 miles, Jain theories 9.2 miles) as it 
was in vedic times, he would have got correct 
value of Sun's diameter. This shows the absolute 
faith of Candrasekhara in ancient scriptures without 
which he never confirmed any result. To some 
extent it was justified, as seen from correct 
assessment of Sun's size in vedas compared to all 
ancient measures. 

Some of his observations may appear biased 
or excessive, but they show a marvelous grasp of 
modern physics. Some of the points were not 
properly understood by top astronomers of his time 

REFERENCES 

1. Bharatiya Jyotisa Sastra - by Sankara Balakrsna 
DIksita - Original written in 1896 in Marathi 
English translation published in 1968 in two 



Introduction * 25 

volumes by Publications division, Govt, of 
India. Translation in English by Sri R.V. 
Vaidya. Hindi translation by Sri Visvanatha 
Jharakhandi published by govt, of U.P. from 
Lucknow. 

2. History of Astronomy in India - Editiors S.N. 
Sen and K.S. Shukla. Published by Indian 
National Science Academy, New Delhi-2 in 
1985. 

3. Indian Astronomy - A source book compiled 
by B.V. Subbarayappa and K.V. Sarma. 
Published by Nehru Centre, Bombay - 18 in 
1985. 

4. History of the calender by M.N Sana and 
N.C. Lahiri Published by Council of Scientific 
& Industrial Research, Rafi Marg, New Delbi-1 
in 1992. 

5. Vedic Astronomy by Sri P.V. Holay, Nagpur 
- 12 in 1989. 

6. Commemoration volume, 1990 - Directorate of 
Culture, Govt of XDrissa. 

* 

7. Introduction to cosmology - Cambridge, 
University, The Structure of the Universe - 
Oxford University Both by Sri Jayanta Visnu 
Narlikara. 

14.4.1997 Arun Kumar Upadhyay 

Ramanavami Cuttack 



MADHYAMADHIKARA 

Madhyama = mean. This portion deals with 
average or mean motion of planets. Calculation of 
mean position is done from beginning of Kalpa or 
yuga for siddhanta, from Kali beginning in a tantra 
and from epoch of this book (12-4-1869 Monday - 
1st saura Caitra 1791 Saka) as a Karana book. 
Siddhanta Darpana explains all the three methods 
and in addition last chapter gives easy method for 
(Calculations of pancanga. There are 33 mathematical 
tables in the end for ease in calculations.. 

The book is in two halves. First half deals 
with the (ganita) methods of astronomy, 2nd half 
deals with explanations and discussions and special 
topics (Gola) 

First half of the book contains three parts 
called Adhikaras. First part is madhyamadhikara, 
with 4 chapters (called Prakasa), Part 2 is 
Spastadhikara with 2 chapters. Part 3 is 
Triprasnadhikara with 9 chapters. 

Second half is called gola and has two parts. 
Part 4 is Goladhikara containing 6 chapters. Part 
5 is Kaladhikara containing 3 chapters. 

First chapter of part 1 is called kala varnana 
explained below. 



Chapter - 1 
MEASUREMENT OF TIME 

1. An Introduction to the Units of measurements 

Any natural science involves theories and 
experiments which verify each other. We test the 
theory by measuring certain quantities and see 
whether they are according to the theory. The 
deviations or errors cause refinement in the theory. 

As in physical sciences (particularly 
mechanics), the units of measurement in astronomy 
are of length, degree and time. Basic units in 
physics are of length, time and mass. Degree is a 
dimensionless quantity because it is ratio of length 
of arc to length of radius. 

Practical units of quantities are based on 
human experience. Length is similar to hand or 
feet length, mass is mass of rice measured by 
spread of palms, time units is based on breathing 
time of human beings. , . 

However, standardisation of length units is 
based on dimension of earth or comparison of some 
light wavelength. Similarly time units are fixed 
according to rotation periods of sun and moon or 
more accurately time taken by light to travel a 
particular distance. We can see that in modern 
physics as well as in ancient India *standardsation 
method was exactly the same. Units of angle also 



25 Siddhanta Darpana 

are based on the number of days (about 360) in a 
year, and hence sexagesimal (divisions by 60) 
system was more convenient. 

2. Units of length 

In British system of units, foot was the basic 
unit equal to average length of human feet. In old 
Greece and Rome; cubit (18" = one hand) and stadia 
were also based on human measurements. For 
smaller units, angula (finger width) was the basic 
unit in India (0.75" or 1.88 cm). 

In Tiloya Pannati (Jain Text), 1 angula = 8 9 
Trasarenu In Anuyogadvara Sutra ("), 1 angula = 

glO // 

In Siddhanta Jyotisa (Sripati), 1 angula = 8 6 
trasarenu 

Successively smaller units of Siddhanta are 

Angula - yava - yuka - liksha - Balagra - Renu 
- Trasarenu. Balagra (hair end) is Angul -s- 8 4 = 1/4 
x 10 4 crri (micron) Thus the dimensions are really 
correct has hair is 3-4 micron wide. 

According to Tiloyapannati lowest division is 
1 paramanu = 1 angula (1.88 cm) x 8* 13 cm. = 3.5 
x» 10" 12 cms. 

This is of the order of nuclear diameter. 

In Lalita vistara (Buddhist text), units are 
divided by 7 at each stage. According to it, 1 
paramanu = 1 angula (1.9 cm) x 7 10 cm = 0.66 x 
10 8 cm. 

This is equal to the Bohr rodius of Hydrogen 
atom. 



Measurement of Time 29 

Larger units are multiples of angula or a 
'purusa' or person (about 6 ft height). It is same 
as 'fathom' used to measure depth of sea or river. 

Bigger units in Tiloyapannati are - 

6 Angula = 1 pada (foot) 

2 pada = 1 vitasti (span) 

2 vitasti = 1 hasta (forearm or cubit) 

2 hasta = 1 rikku or kisku 

2 Kisku = 1 danda (staff) or dhanusa 

2,000 danda= 1 Krosa 

4 Krosa = 1 yojana 

Same units have been used by Paulish 
Siddhanta, Srlpati and subsequent siddhanta texts. 
Lalita vistara, however makes 1 kosa = 1000 
dhanusa only equal to 1/2 Jain or Siddhanta yojana. 

In the time of Napolean, attempt was made 
to link length unit 'metre' with dimensions of earth. 
So 1 metre was proposed to be 10" 7 of distance 
between equator and north pole. Subsequently, it 
was learnt that it was 1 crore and 486 parts of this 
distance. Still, the standard length of platinum bar 
kept at Paris is used as metre. Nautical mile is also 
based on earth's dimension but it is not a 

decimal fraction. It is length of 1 minute of arc at 
equator (about 6080 ft. or 2 kms) 

In same way yojana has been defined to be 
an exact fraction of earth's diameter or circum- 
ference in polar circle. 

Varahamihira - Circumference 3200 yojana 

Aryabhata - Diameter = 1050 yojana 

«■■ 
Surya Siddhanta - Diameter = 1600 yojana 



30 Siddhanta Darpana 

Siddhanta Siromani - Circumference = 4,800 
yojana 

(This is followed by Siddhanta Darpana also) 

Thus, yojana is 5 miles according to Siddhanta 
Shiromani and 7.52 miles according to Aryabhata. 

Anuyogadvara Sutra (Jain) gives 

1 Atma yojana = 7,68,000 angula = 9-1/11 miles 
estimated according to current measurements of 
earth. Dr. L.C. Jain opines that 1 Pramana yojana 
is 500 Atma yojana = 4,5 45.45 miles. M.B. Panta 
opines that 5 yojana (40 or 45.5 miles) was called 
Mahayojana used for measuring distances of stars. 

For example, 'Trisanku' star is named on basis 
of its distance from earth. 

Tri-Sanku = 3 x 10 13 Mahayojana = 207 light 
years. This is actually the distance of that star now 
known as Beta-crucis in Southern cross constella- 
tion. 

Similarly, it is said that Agastya had crossed 
Varidhi (10 14 ) or drunk ocean and had gone south. 
It is now known as 'Argo-Navis' star at 80° 5' south 
latitude, indicating naval journey. This star is 652 
light years away ; 10 14 mahayojan is about 690 light 
years. 

Prof. S.S. Dey of Calcutta has observed that 
Egyptian names of planets mercury, venus, mars, 
jupiter and saturn give their distances from Sun 
in yojana if names are interprated in Katapayadi 
system. 

At present metre is defined as 16,50,763.73 
times the wave length of radiation of Krypton-86 
isotope for transfer of electron between 2p and 
4 d states. With accurate measurement of velocity 



Measurement of Time 31 

of light, it is proposed to link time and length 
units. In fact truti (a unit of time) was also defined 
as time taken by light to travel 1 ybjana. 

3. Measurement of Time 

Principle of time measurement is to choose a 
unit which is equal to the time of a periodic event 
(which repeats itself after fixed intervals). Examples 
of such events are - vibration of quartz crystal or 
metal spring, pendulum (all used in clocks), 
rotation of earth (l^day = 24 hours), synodic 
rotation of Moon (1 month) or apparent rotation 
of Sun around earth (1 year). 

Basic unit of time in jyotisa is 'asu' (meaning 
mouth) or 'prana' (breathing) as it is approximately 
time (4 seconds) taken by a man in breathing in 
and out. Since our mental feeling of time is based 
on breathing only, units bigger than asu can be 
felt and are called 'Murtta' (tangible). Smaller units 
are called Amurtta (imaginary). Astronomically, it 
is time taken by earth in its daily motion (360° in 
24 hours) to move by 1' (=1760). 

'Amurtta' or small units - In ganita-Sara- 
Samgraha (Jaina) 1 prana has been divided into 
44466-2458/3773 Avalikas. Possible reason for such 
peculiar ratio is that a muhurtta (1/30 of a day of 
24 hours) was equal to 3773 pranas in one system 
and 1,67,77,216 avalikas in another system. 

A solar day is divided into 60 danda or ghatika 
(like 24 hours). Each ghati is divided into 60 pala 
(each 24 seconds) which is again divided into 60 
vipalas. 

Thus 1 asu or prana is equal to 1/6 pala or 
10 vipalas 



32 Siddhanta Darpana 

1 asu (respiration) = 5/2 kastha 

1 kastha = 4 long syllable (gurvaksara) 

1 gurvaksara or vipala = 9/2 nimesa (twinkling 
of eye) 

1 nimesa = 100 lava 

1 lava = 100 truti (1 Truti is time taken by a 
sharp needle to pierce a soft lotus petal) 

1 Truti = 3 Trasarenu 

1 Trasarenu = 3 ami 

1 Ami = 2 paramanu 

TU - T ^ 1 asu (4 second) 

Thus 1 Truti = — — * — — - — 

10 x 9 x 50 x 100 

In this time, hight will travel about 2.68 Kms 
(1/3 or 1/4 yojans or 1 krosa approximately). 

1 paramaru Kala = 1/8 Truti = 5 x 10" 7 seconds 
approx. 

Larger Units - 

10 gurvaksara or Vipala = 1 prana 

6 prana = 1 pala or vighati 

60 vighati = 1 ghatika 

60 ghatika or danda = 1 day (24 hours) 

30 days = 1 month (approximate time from 
one full moon to the next) 

12 months = 1 year (approximate time of 
apparent rotation of Sun) 

360 years = 1 divya varsa (divine year) 

43,20,000 years = 1 yuga 

72 yugas = 1 manu 

14 Manu = 1 Kalpa (day of Brahma) Aryabhata 



Measurement of Time 33 

Thus in this system 1008 yugas make a kalpa. 

Suryasiddhanta gives 1000 yugas in a kalpa 
with 14 manus of 71 yugas each with 15 sandhis 
of 1 satyuga (4/10 yuga) each. 

2 kalpa = 1 ahoratra (day-night) of Brahma 
30 days of Brahma = 1 month of Brahma 
12 months of Brahma = 1 year of Brahma 
= 7,25,760 yugas (Aryabhata) 
or 7,20,000 yugas (Surya Siddhanta) 

100 years of Brahma = Life of Brahma 
(Mahakalpa or Para) 

50 years is called Pararddha = 1.5 x 10 17 years 
In one mahayuga there are 0.4xl0 17 asus 
Hence 10 17 is called para or parardha. 

Concept of yuga - 'Yuga' of Rkveda was of 
19 years after when mutual motion of moon and 
sun repeats itself. Later on, this period was called 
metonic cycle in Greece. Yajur jyotisa gave a yuga 
of 5 years which is a simpler system of tallying 
lunar and solar years. In vedanga jyotisa 19x8+8 
= 160 years was next bigger yuga after which 
lunisolar calender tallies more accurately. 
Visvamitra had smaller yuga of 3339 tithis = 111 
synodic months + 9 tithis. This was half of Saros 
cycle of Chaldea (223 synodic months or 18 tropical 
years and 10.5 days) after which ellipses are 
repeated. His greater yuga was of 3339 synodic 
years or 3240 sidereal years. One third of this 
period 108(0 sidereal years was used in determining 
Indian Eras. This gave rise to small chaturyuga of 
4x1080 = 432p years. One Mahayuga is JL0O0 times 
this unit and 1 Kalpa is 1000 mahayuga. This is 



54 Siddhdnta Darpana 

based on astronomical hymn of Visvamitra (RV III 
9-9) - 3339 dyus (days/ tithis/parts of sky) wor- 
shipped Agni (Sun) by revolutions in the sky. This 
concept has been used for divisions of constellation 
in vedanga jyotisa. 

Siddhanta texts have formed a Mahayuga in 
which all the seven planets Sun and Moon and 5 
faint (Tara) planets make complete revolutions. 
After a yuga they come to the same position. Thus 
a position of these planets will occur only once in 
a yuga and is most accurate method of indicating 
a time in a yuga. This is one of the purposes of 
preparing a horoscope. 

Rotation of mandocca (apogee) of planets is 
still slower and their full rotations are completed 
only after 1000 yugas or a Kalpa. Slowest is sani 
whose mandocca makes only 39 rotations in a 
kalpa. 

It may be mentioned that a period of kalpa 
of 4 bilion years is approximately same as life of 
earth or the solar system. 2 kalpa or 1 day/night 
of Brahma is considered to be the time from when 
universe is expanding and will contract again. Life 
of Brahma 3xl0 17 years is approximately half life 
period of proton decay^after which basic elements 
of the universe will dissolve themselves. 

4. Other examples of accurate measurements - 

Verses of Veda composed by known 
astronomers Visvamitra, Atri, Sunahsepa, 
Hiranyastupa, Kutsa, Utathya, his son 
Dirghatamas, his son Kaksivat and daughter Ghosa 
- should be read according to Katapayadi system 
for their mathematical meaning. 



Measurement of Time 35 

Nasadlya and other verses of these sages 
indicate theories of creation of universe which are 
similar to modern cosmology. 

Rkveda (1-164-2) tells that the seven join the 
body in constant circular motion of earth (ratham). 
Orbit round Sun is elliptical (called Trinabhicakram) 
because elipse has 3 nabhis (1 centre and 2 focus) 

Cakram = 2x^= 6.283, ratha = 72 

Hence in Krosa units (= 2.5 miles) 

7 x Cakra x ratha = 6.283x7x72x2.5 = 7915 
miles which is diameter of earth. 

Second line indicates Sapta (7) nama (50) 
vahati (moves in orbit). If movement is taken in 1 
lava 

muhurta 

= — = 48 seconds. 

60 

then drbital velocity of earth is 

7 x 50 Krosa 7 x 50 2.5 miles 

1 lava 48 seconds 

miles/sec. 

Tri (3) nabhi (40) Cakra (2rr = 6.283) gives 
acceleration due to gravity if length unit is taken 
as hasta 19.8" and time as lava. For small units 
both are divided by 60. 

Vilava = 4/5 sec, 1/60 hasta = 0.33/12 ft. 



0.33 
g = 3 x 40 6.283 x —— x 



4 



2 



= 32 ft/sec 2 



Fourth line gives arc of an imaginary sphere 
on which moon moves. 



36 Siddhdnta Darpana 

Yatra (21), visva (44), Bhuvana (44) gives 
21x44x44 = 40, 656 when unit is moon's distance 
-s- radius of earth. DIrghatamas gives a theory of 
star formation in RV (1-164-8) - Mata (steller cloud 
formed by hydrogen atoms) absorbs light 
(garbharasa) and is further excited by gravilational 
contraction (pitaram). Dhiti (69) manasa (708) gives 
diameter of hydrogen atom if we take unit of length 
60x60 times smaller than 1/60 hasta (0.33 inches) 

Dhiti x manasa 

0.33 1 

X rrr X 2.54 cm. 



60 X 60 60 x 708 

= 0.478 x 10" 9 cm = radius of hydrogen atom 
when it is divided by mata pitaram (65x261), this 
gives 2.8xl0* 13 cm, the distance at which nuclear 
interaction works. Atomic radius divided by Sa (7) 
garbharasa (7243) gives 10" 13 cm which is diameter 
of proton or electron. 

Velocity of light - Smrtisastra tells - we salute 
with our respect to sun who traverses 2202 yojans 
in 1/2 nirhisa. 

In purana - 1 nimisa = 16/75 seconds 

In Luavatl, 1 yojana = 4x8000 cubits = 9.09 
miles 

Hence velocity of light is 

9.09 x 2202 „ ^ . ._ , 
= 1.86 x 10 5 miles/sec. 

8/75 

Bhaskara nimisa is 8/90 seconds, Manu's 
yojana is 4 Krosa of 4000 cubits each. Then velocity 
is 3xl0 5 km/sec. 

5. Measurement of angles - Since apparent 
revolution of Sun around earth in a year is in 
about 360 days, a circle is divided into 360 ° degrees 



Measurement of Time 37 

(amsa) so that motion in one day is about 1°. Its 
average motion in 1 month (lunar node to node) 
is about 30° hence 30° is 1 rasi. Further sub 
divisions are always by 60 at each step because it 
is a simple factor of 360 and there were 6 days 
weak (sadaha) in vedas. 1 extra day was added to 
some weeks making it 7 days, this day was not 
regular weak day hence tradition of weekly holiday 
arose. Since moon's node makes 12 rounds when 
Sun makes 1 round, the clock also copies that 
motion. Minute hand makes 12 rounds when hour 
hand makes 1 round. 

Angular and time measurement both are 
divided into 60 units so that they tally with sun's 
motion. 

One rotation = 12 rasi = 360 Arhsa (degree) 

1 Arhsa = 60 Kala (minute) 

1 Kala = 60 vikala (second) 

Tatpara and paratpara are further divisions. 

Thus angular motion of sun corresponds to 

time units 

1 rasi = 1 month, 1° = 1 day 

1' = 1 danda, 1" = 1 pala 

1 tatpara = 1 vipala etc. 

This division of angle continues throughout 
the world till day. Time units are slightly different, 
but still in divisions of 60. Actually hour is derived 
from 'Hora' (Ahoratra) i.e. two divisions of a rasi 
(like day-night divisions of a day). Earth rotates 1 
circle i.e. 24 hours in 1 day, hence 1 hora in 1 
hour. 



38 Siddhanta Darpana 

6. References to introduction 

(1) For units and dimensions any standard 
text book of physics for +2 or graduate standard 
may be referred. First chapter is on units and 
dimensions. 

(2) For vedic astronomy see P.V. Holay's book. 

(3) For other interpratation of vedas see - 
Issues in vedic Astronomy and Astrology published 
in 1992 by Rashtriya Veda Vidya Pratishthan, New 
Delhi-2. 

(4) For details of siddhanta texts individual 
texts may be referred. Some information is com- 
piled in chapter 7 of Indian Astronomy - A source 
book. 

Translation of the text (Chapter I) 

Verses 1-9 - Mangalacarana - Prayer to Lord 
Jagannatha and other gods. 

Verses 10-11 - Pratijna - Scope and purpose 
of the book - It deals with only ganita jyotisa. 
Purpose is to explain difficult methods of mathe- 
matics in simple language to a common man. 

Verse 12 - Mathematical methods (Patiganita) 
has already been perfectly explained by Sri 
Bhaskaracarya in his text Lflavaa* . Without repeat- 
ing the same, motion of planets is discussed 
straight away. 

Verse 13 - Comparison with earlier Siddhantas 
- Some special subjects have been dealt with in 
this siddhanta, not found in earlier texts, for 
satisfaction of the learned. 

Verses 14-15-Importance of jyotisa - Veda 
guides everyone in yajna. Muhurta (auspicious 



Measurement of Time 39 

time), for that is known through jyotisa. If veda 
is taken in human form, Jyotisa is eyes, Vyakarana 
is mouth (grammer), Nirukta (dictionary) is ears, 
kalpa (Purana) is hands, Siksa Sastra (reading) is 
nose, chanda (prosody) is feet. Thus jyotis is the 
part of veda through which all other parts can be 
understood. 

Comments - Many portions of vedas, Samhitas 
and all the text books explain jyotisa in similar 
words which need not be repeated. 

Verses 16-19 - About Siddhanta - Without 
mathematical astronomy, the whole jyotisa is 
ussless. Siddhanta deals with time scale from Truti 
(smallest unit) to Kalpa (biggest unit used in 
jyotisa), arithmatics (including indeterminate equa- 
tions of first and second degree), algebra, evolution 
and creation of world, orbits of earth, planets and 
stars, eclipse and conjunction of planets and 
description of various instruments like jya, dhanu 
etc. and elements of mensuration. Among all 
sastras jyotisa is highest; in jyotisa itself, siddhanta 
is best; and in siddhanta also gola (= sphere 
including Bhugola = geography and khagola = 
astronomy) is most important. A country prospers 
due to presence of men well versed in gola. 
Otherwise, animal behaviour 'spreads. Siddhanta 
gives all the four results (Dharma, Artha, Kama 
and Moksa). So Surya (creator of Surya siddhanta) 
has kept it a secret to be given only to good and 
pious prson. 

Verse 20 - First half of this text deals with 
time units, ahargana (count of days), bhagana 
(revolutions in sky), graha anayana (calculating 
planetary positions), jya (sine) kotijya (cosine) etc. 



40 Siddhanta Darpana 

spasta sara (actual position of planets as seen in 
the sky), triprasna (three problems about daily 
motion). 

Second half deals with different theories, 
creation and dissolution (srsti and laya), earth 
(geography), kaksa (orbit), yantra (instruments), 
description of countries, prayers to lord Jagannatha 
and Kautuka Panjika (easy preparation of almanc). 

Verse 21 - Parabrahma was in the beginning. 
It created purusa and prakrti (2), mahattatva 
(intellect-1), ahankara (ego-1), tanmatra (elements 
5), mahabhuta (5 types of creations or beings), 5 
organs of sense, 5 organs of action, and one mind 
- a total of 25 elements. It supervises all. 

Verse 22 - Jyotisa Cakra - After creation, 
Brahma caused the sphere of akasa to rotate from 
east to west in a daily motion (seems as a result 
of rotation of earth on its axis from west to east) 
with respect to two dhruvas (north and south poles 
on the axis). With a slower motion, the planets 
move west to east relative to stars in nica and ucca 
circles (earth is not the centre of their or bits). 

Verses 23-30 - Kala has two meanings - one 
is destroyer of world and the other is reckoning 
of time. Time units are of two types - Suksma or 
amiirtta is very small unit which cannot be felt by 
senses, but calculated or measured by instruments. 

Prana is of 4 seconds, in which a person 
breathes in and out. This is the smallest sthula or 
murta time unit to be felt by senses. 

Divisions of time are - 1 lava = 100 truti 

30 lava =1 nimesa = 1/135 seconds 

18 nimesa = 1 kastha 



Measurement of Time 41 

27 nimesa ■ Time to pronounce long vowel 

=0.2 seconds 
20 long vowels =1 prana (4 seconds) 

2 Pranas =1 Kala 

3 Kala 1 vighatika or pala (24 seconds) 
10 vighafi = 1 ksana 

6 ksana = 1 ghati or danda (24 minutes) 

2 danda 1 muhurta 

* 

30 muhurta = 1 naksatra dina 

Note - Naksatra dina is the time of rotation 
of earth with respect of stars (23 hours 56 minutes) 
called sidereal day and is slightly smaller than civil 
day or solar day (between sunrise to sunrise) of 
average 24 hours. 

30 naksatra day = 1 naksatra masa 

From sunrise to next sunrise is called savana 
dina (or civil day) 

30 Savana dina = 1 savana masa 

30 tithis = 1 candramasa 

NB-1 Candramasa is period from moon's node 
(amavasya - in same direction as sun, or Purnima, 
at 180° from Sun) to the same node next time. 
Tithi is l/30th part of that period equal to the time 
in which moon gains 12° difference over sun. 

Verse 31 - Detailed description of these 
naksatra and civil days, months etc. will be done 
in second part of the book. 

12 solar months = 1 solar year 

Solar (Saura) year is one day of deva or asura 
(divya or Asura varsa) 

Note - 1. Sauradina is the interval of time 
during which sun moves 1° of ecliptic. In saura 



42 



Siddhanta Darpana 

masa it moves 30° or 1 rasi and in 1 year it makes 
a complete round of 360°. 

2. Krantivrtta or ecliptic is the apparent path 
of stars from east to west in plane of equator. One 
complete round is called bhagana which is divided 
into 360° amsa (degrees). Each subdivision is in 
60's as follows - 

60 viliptas, vikalas, or seconds 

= 1 lipta, kala or minutes 

60 liptas = 1 amsa, Bhaga or degree 

30 amsa = 1 rasi or sign 

12 rasis = 1 Bhagana or revolution 

Verse 32 - When sun is moving north of 
ecliptic (for six months), it is day for devas and 
night for asuras (day in north pole and night in 
south pole). When sun is moving south of ecliptic, 
it is night for devas and day for asuras. 

360 divya or asura days = 1 divya or 
asuravarsa 

Verse 33-39 - Time scales greater than a year 

Period Divya years Solar Years 

Satya yuga 4,800 17,28,000 

Treta yuga 3,600 12,96,000 

Dvapara yuga 2,400 8,64,000 

Kali yuga 1,200 4,32,000 

Total - yuga 12000 43,20,000 

or Mahayuga 

There is a sandhya, l/12th of the yuga, 
included in each yuga at its beginning and the 
end. Total sandhya is 1/6 of the yuga. 



Measurement of Time 43 

Sandhya at beginning Divya years Solar years 

or end 
Satya yuga 400 1,44,000 

Treta yuga 300 1,08,000 

Dvapara yuga 200 72,000 

Kali yuga 100 36,000 

1 day of Brahma is called kalpa and it consists 
of 1,000 yuga. Kalpa is divided into 14 manvantaras 
of 71 yuga each. They are separated by 15 sandhya 
periods in between and at the end, in addition to 
manu period. Each sandhya is equal to one satya 
yuga i.e. 4/10 of a yuga. 

Thus, 1 kalpa = 14 manu + 15 sandhya 
= 14 x 71 yuga + 15 x 4/10 yuga 
= 994 yuga + 6 yuga = 1000 yuga 
Note - According to Aryabhata, a kalpa has 
1008 yuga divided into 14 manus of 72 yugas each. 
Verse 40-46 - Current time - At present 50 
years of Brahma have passed. In the 51st year (of 
2nd parardha or half life of Brahma), this is the 
first day, called Svetavaraha kalpa. In this kalpa, 
six manvantaras have passed, namely - (1) 
Svayambhuva (2) Svarocisa (3) Auttami (4) Tamasa 
(5) Raivata and (6) Caksusa 

Current manvantara is Vaivasvata in which 27 
yugas have passed. In 28th yuga, Satya yuga, Treta 
and Dvapara have gone. First fourth part of kali 
era is continuing. 

Time from beginning of creation till beginning 
of kaliyuga in this kalpa - 

Beginning sandhya " = 17,28,000 years 

6 Manvantaras = 6x71x43,20,000 = 1,84,03,^0,000 years 



44 Siddhdnta Darpana 

6 sandhyas of 6 manus 6x17,28,000 = 1,03,68,000 years 
27 yugas in 7th manu 27x43,20,000 = 11,66,40,000 Years 
Satya, Treta, Dvapara = 38,88,000 years 

Total = 1,97,29,44,000 years 

In kaliyuga, as in March 1996, 5098 years have 
passed. These are to be added to find the years 
passed since beginning of kalpa. 

Verse 47-52 - Motion of planets started at mid 
night at Lanka (a point of equator through which 
prime meridian through Ujjain passes). The day 
was named as Ravivara, Caitra Sukla pratipada. 
All the planets reach the same position at midnight 
of Brahma (after interval of 1 kalpa) 

At the end of Brahma's day (kalpa) all planets 
vanish. Author doesn't agree with Bhaskara that 
earth remains. 

Verse 53 - Bhagana is one revolution of a 
planet starting from Asvini to Revati end as seen 
in the sky. 

Explaination - Path of revolution along Zodiac 
(apparent path of planetary movement, more 
correctly of sun - rasivrtta) covers 360° divided in 
12 rasis of 30° each. Almost the same circle is path 
of moon (inclined at 5° angle) There are 27 
naksatras of 13° 20' each in which moon stays for 
about 1 day each. There was system of unequal 
division of naksatras also which will be discussed 
later on. Asvini naksatra is 1st and mesa rasi also 
start from 0° of zodiac. Last nakstra is Revati 

Verse 54 - Division of angular measurements 
according to previous acaryas - 

1 Bhagana 12 rasi, 1 rasi = 30 Amsa 



Measurement of Time 45 

1 Amsa = 60 kala, 1 kala = 60 vikala 

1 vikala = 60 para 1 para = 60 vipara 

Note - Para and vipara are not used by other 
texts nor in modern mathematics. 

Verse 55 - Prayer of Lord Jagannatha 

Verse 56 - Sri Candrasekhara has written this 
drksiddha ganita (as observed in sky) in simple 
language, so that even children can follow it. 




Units of Time and Length -written and added later 
Units of measurement-A physical quantity is measured by 2 components- 
A basic unit of quantity used as standard for comparison of other quantities. 

A number which is ratio of measured quantity to standard .Basic unit should be available, reproduci- 
ble, convenient to handle and easy to compare through experiments. 

How many units are sufficient-For mechanics, 3 units are sufficient-length, mass, time. All other 
units of measurement can be derived from them. In 1901, Giorgi proved that that by adding a quan- 
tity related to electric properties, all physical quantities can be measured. To show the inter-relation 
between electricity and magnetism, or to explain the property of medium (vacuum or material), an- 
other quantity is required. Thus 5 quantities are sufficient to explain all physical quantities. These 
are called 5 Tanmatras in Sankhyg philosophy. For this 5 dimensional view of world, there are 5x5 
= 25 elements in Sankhya. 

5 fold division of units is described by 5 Ma (=to measure) chhandas- 
Ma, pra-ma, prati-ma, upa-ma, sa-ma 
For same type of quantity e.g. length- 

Ma- basic unit (e.g. meter), Pra-ma = multiples (kilometer etc.), Prati-ma - sub-multiples, Upa- 
ma = Related length units (Foot, Nautical mile, light-year etc.), Sa-ma = Link with other units (with 
time through velocity of light, or with area, volume etc.) 

Inter-relation of units of same or different kinds is called asn-vaya (upama + sama) 
Various types of inter-relations are called Vaya-chhandas. In context of length, they are classified as 
-.Ma =Prthvf (earth) -Standard rod, earth or earth-like compact body-sun, solar system, galaxy. 
Prama- Antariksa- Intermediate. Regions between and beyond earth(s). 
Pratima -Space, volume. 
/4s/7i/ADirections. 
Modern units of length-(1) Foot- Based on human foot- 

(2) Meter-It was defined in 4 ways-(i) Length of pendulum with half time-period of 1 second, (ii) 10 7 
part of arc length from equator to north pole, (iii) 16,50,763.73 times wavelength of Kr 8 e radiation 
between energy states of 2 P 10 and 5 d 5, (iv) Distance traveled by light in 29,97,92,458 part of 1 sec- 
ond. 

(3) Nautical mile-1 minute arc of equator. 

(4) Astronomical unit (AU)-Semi-major axis of earth orbit around sun. 

(5) Persec -Distance at which AU subtends an angle of 1 second. 

(6) Light year-Distance traveled by light in 1 year=1016 meters approx. 
1 Persec=3.26 light years, 1 AU=1.5x1011 meters. 

Seven yojanas-{\) Nara yojana =32,000 hands-used for human possessions of land. 

(2) BhO-yojana — 1000 or 1600 parts of earth diameter (surya-siddhanta -8km-yq/ana) 

(3) Bha-Yojana — 27 bhu-yojanas =216 kms, used for sun distance and size of galaxy 



(4) Prakasa-yojana — Distance traveled by light in 1 truti- 1/33,750 seconds 

(5) Dhama-yojana-Ksara dhama =720 parts of equator circumference = 55.5 kms. 

Aksara dhama-Measure of space with earth as standard in exponential scale. Number of powers of 
2 is equal to akshara in chhanda. Distance d = r x 2 ( n - 3 ), r = radius of earth, n = unit of distance 

(6) Sun diameter as yojana for solar system in puranas. 

(7) Pramana-yojana — Starting from solar system, scale for each successive /oka in 500 times longer 
units at each step. 

Micro-units-Smaller worlds are successively 1 lakh times smaller-man (meter size), kalila (cell, 1 
lakh part of meter), JTva (atom of 10(- 1 °) meter size), KundalinT (nucleus of 10 15 meter size), Jagat- 
particles of 3 types (10 20 meter size-not defined- Chara (lepton), sthanu (Baryon), AnupOrvasah (link 
-particles or meson) 

Deva-danava (10 25 meters size)-Creation from 33 types of devas only, not from 99 types of asuras 
(danava), Created world is 1/4 th part of purusa. 3/4 th is field or dark matter. 

P/fe/^proto-type, Parents)-10- 30 meter size. 

Rsi (string-rass/in hind/) 10 35 meters size. 

Micro or smallest unit called paramanu by Varahamihira = 8 4 parts of angula = 4.5 micron. Snpati 
calls it trasarenu equal to 60 atoms. So, atom =1.2 x 10 7 cm. 

Lalita-vistara-Paramanu-raja = Angula x 7 10 = 0.6 x 1 8 cm 

Tiloya pannati— Trasarenu = Angula x 8 9 = 1 .4 x1 8 cm. 

Single object or Brahma is indicated by Angustha (Thumb). In Purusa-sukta, angula means 96 parts 
of human length, or earth, solar system earth, galaxy as per context. 

Measures of Solar system-Modern estimate (NASA 2002) is that Woort cloud is boundary of solar- 
system at distance of 50,000-1,00,000 AU from sun. Indian measures- 

Samvatsara is aditya (energy-field) of sun. Fields of galaxy and universe are called Varuna 
and Aryama. This is sphere of 1 light year radius with center at sun. 

1,575 crores diameter in unit of sun-diameter = 2.310 light year.(called ratha of sun). Outer 
wheel diameter is 6,000 i.e. outer boundary of Kuiper-belt. Modern estimate is about 70,000 
plutonic bodies of above 100 km size. Puranas tell 60,000 balakhilyas of Angustha size i.e. 
96 parts of earth diameter =135 km approx. Surya-siddhanta calls it Naksatra-kaksa of sun at 
60 AU. This has been called A/oka (dark) earth of 100 crore yojanas (8km). Loka (lighted) 
earth is of 50 crore yojana (8 km) diameter. This has 7 dvTpa /oceans of Priyavrata formed 
by motion of planets. Inner wheel of sun is of 3000 sun diameter i.e. up to Uranus orbit. Prac- 
tical or Indra zone is of 1000 sun diameter {sahasraksa, aksa = sun or eye) up to Saturn orbit. 



Saturn being at the end of solar effect is called son of sun. 

Planetary distances in Bhuvar-loka are in terms of earth diameter. Size of next Svar-loka will be 
in 500 times bigger unit. Distance from sun to pole {Dhruva) is 14 lakh x 500 earth diameter 
which is distance of Woort cloud. 

Earth diameter x 2( 3 °) in aksara-dhama units. With three spherical zones inside earth as image of 
2 bigger earths, there are 33 zones. Energy (prana) of each is a deva. Signs of 33 devas are 
letters from k to h - its arrangement (nagara) is Indian scripts called Devanagarl 

Earth/man = Solar earth/Earth. =10 7 , called koti (limit). This is called Maitreya- mandala or SavitrT 
(2 24 of earth size) in which creation occurs. Dyu (sky) of solar system is 10 7 times sun size. 
Earth is crore times or 24 dhamas bigger than man and is called GayatrT SavitrT x 2 24 = 
SarasvatT (creative field of galaxy). SarasvatTx 2 24 = Veda (creative field of universe or Veda- 
purusa^O times bigger than universe). Creative aspect is Niyati. 

Measures of Galaxy-(1) SOrya-siddhanta gives 1.87 x 10 16 Bha-yojanas (216 km) = 1,23,000 light 
years (modern estimate 1 lakh LY) 

(2) Earth size x 2 46 is galaxy Its creative field is KOrma (Go/oka of Brahmavaivarta purana) of 52 
dhama units i.e.2 49 earth size. For 49 ahargana (dhama units) there are 49 letters in Devanagarl 
script from a to h. 3 extra units of KOrma is creator, conscious being is called ksetrajna in GTta chap- 
ter 13, so 3 letters are added at end-ksa, trajha. 

(3) Circumference is 0.5 Para (10™) dhama yojanas (55.5 kms). Diameter comes to about 1 lakh LY. 

(4) Size of KOrma in Narapati-Jayacharya\s hundred thousand (10 5 ) Sanku^O™) = 10 18 yojanas. As 
a purusaof Galaxy, it is 10 times bigger, so galaxy is about 10 17 yojanas. In space, earth is lotus of 
1000 petals, 1 petal = 1 yojana (Aryabhata) 

(5) Galaxy is 1 crore times solar earth- SavitrT. 

(6) This is Janah loka of 2 crore yojana radius C\ yojana = 500 x 500 sun diameter) 

Bhuvar-loka is sphere of 15 ahargana {dhama) around earth i.e. 2 12 of earth size which is also called 
Varaha. Viewed from sun, it is 100 yojana ( = sun diameter ) high and 10 yojana high, up to lunar 
orbit. This covers up to 60% distance of Venus orbit. Earth's exclusive zone extends up to 9 
ahargana i.e. 64 times earth radius. Moon is within it at 61 radius. Spiral arm of galaxy is called 
Sesa-naga. Near sun, it has 1000 sun like stars called 1000 heads of Sesa. Region of this is called 
Maharloka, whose size is given as- 

1000 times size of solar system (sahasra-sTrsa purusa,\ 000 heads of Sesa) 

43 ahargana =earth x 2 40 (tristup chhanda has 44 ± 2 or 43 letters in Mahesvara sOtra) 

1 crore yojanas (1 yojana =500 sun diameters) 



Middle loka between earth (Jbhu) and satya loka 10 20 times bigger, so it is 10 10 times earth size. 
Tapah loka is given in 4 ways- 

2 64 times earth size for 64 letters in BrahmJ script. 
8 crore yojana (1 yojana =sun diameter x 500 x 500) 

(3) 864 crore light year radius -equal to day-night of Brahma. 

(4) Earth orbit/earth = Tapah loka/ga\axy 
Satya loka (1) Galaxy x 10 7 (or 2 24 ) 

(2) Maharloka /earth = Satya loka / Mahar 

(3) 12 crore yojana (1 yojana = sun diameter x 500 3 ) 

Time units 

Modern unit-1 second = 86,400 parts of mean solar day. 

Due to fluctuations and slowing down of earth rotation by tidal friction, new definition was adopted 
in1967-it is 9,19,26,31,770 times the period of light radiated by transition between two ground states 
of Cesium-133 atom. 

Nine Indian Time-units- 

(1)Brahma-T\me period of creation from formless {avyakta) to forms is called a day of Brahma or 
Kalpa. In same period of time, creation dissolves into avyakta {GTta 8/17,18). Day of Brahma 
has been defined as of 1000 yugas in GTta, puranas, Surya-siddhanta etc. Each yuga is of 
12,000 divya-years, 1 divya-year =360 solar years. Thus, 1 day of Brahma = 432 crore 
years. 

(2) Prajaptya-Prajapati started yajha {GTta 3/10), so prajapatya period is period of galaxy from 
where creation started. Rotation period of galaxy is called Manvantara of about 31.68 crore 
years. Present stage is 7 th manvantara. This has been called 7 th day in Bible. After 4, 5 th days 
sun, moon, earth were created. So, day cannot mean here rotation of earth or even of sun. 
This is rotation of first creation galaxy. Modern estimate of period of sun revolution around 
center of galaxy is 20-25 crore years. 

(3)Divya year is of 360 solar years, arrived in 3 ways. This is approximate period of revolution of 
imaginary planet at 60 AU (or average rotation of about 60,000 Balakhilyas at same dis- 
tance), called A/oka (dark) boundary of 100 crore yojana diameter, This has been called pari- 
varta yuga in Vayu, Matsya puranas, which is cycle of historic changes. Third view is that 
north-south motion of sun is like day-night cycle. This cycle of 1 year is 1day (d/vya). Taking 
round number 360 for days in a year, d/vya year is of 360 years. 



(4) Guru scale-In period of 60 years Saturn and Jupiter complete integral revolutions-2 and 5. 
Alternatively, Ang/ra-effeci (upward convection due to radiation pressure takes 60 years to 
complete {Aitareya Brahmana 18/3/17, Taittinya Brahmana, 2/2/3/5-6) 

(5) Pitar mana- Synodic revolution of moon in 29.5 days is called 1day of pitars. Varaha is 15 ahargana or 4,096 
times earth size-that is parjanya. Intermediate level is pitar, 64 times earth size called pitar. Moon orbit is 61 
times earth size. So pitars of human beings also reside on outer region of moon. Our bright half of month is 
night of pitars and dark half is their day. 

(6) Savana mana-Sunnse to next sunrise is savana or civil (practical) day.1 month = 30 days, 1 year = 12 months. 

(7) Solar-Apparent revolution of sun around earth is yea r(sidereal). 1/12 th part is 1 month (1 rasi = 30°. Movement). 

1 day is 1° movement. 

(8) Lunar-Synodic revolution of moon around earth is a lunar month. 5 angas of panchanga are defined from moon 
and sun — Tithi- (M-S) /12°, \tithi- 2 karana, yoga = (M+S) /13.3. Naksatra is average daily motion of moon. 
Vara (day) is cyclic naming of days. 

(9) Nakstra (sidereal)-Axial rotation of earth with respect to fixed stars is sidereal day of 23 hours 56 minutes. Month, 
year have 30 and 360 days. 

Micro units of t\me-Satapatha Brahmana (12/3/2/1,5) divides mean day length of 12 hours successively by 15 parts into 
units named as muhurtta (48 minutes), ksipra, etarhi, idanT, prana, aktana (or, ana), nimesa, lomagartta, svedayana. It is 
stated that stars (naksatra) in galaxy (Brahmanda) are like its loma-gartta (roots of skin hairs). Number of lomagartta in a 
year (Samvatsara) is equal to the number of stars in a galaxy, so this unit of time is called lomagartta which is equal to 
muhurtta x 15 7 = about 80.000 parts of a second. Similarly its 15 th part is svedayana equal to about 11, 20,000 parts of 
a second. In this time, light travels about 270 meters. Rain drops (sveda) move (ayana) about same distance without 
breaking or joining, so this time unit is called svedayana. Estimate of number of stars in galaxy was done after 1985 
which is correctly estimated in Satapatha-Brahmana to be 10< 11 ). Logic of division by 15 is given that ratio of earth orbit 
to earth size is same as ratio of Tapah-loka (visible universe) and galaxy, both equal to 2 < 15 ). 

Seven Yugas-By joining two cycles of time, a yuga is formed. MunTsvara in his astronomy text Siddhanta Sarvabhauma 
has stated 5 yugas- 

5 years, 5 x 12 = 60, 12 x 60 = 720, 600 x 720 =kaliyuga, kalix 10=1 yuga. 

Like 7 yojanas, there are 7 yugas, depending on completion of various yajhas — 

(1) Sanskara yuga — Education and other reforms projects are completed in 4 to 19 years which is a sanskara-yuga. 
(a) Gopada-yuga-\s of 4 years like modern leap year system. Its year starts in godhuli-vela (literally cow-dust- 
time, when cows return home at sunset, dust is raised) like Hebrew or Islamic calendars. West Asia was place 
of Asuras, called Nisacharas because their day started with sunset.This is described in Aitareya Brahmana 
(7/13). Suppose, 1 st year starts at 6 PM on 4-1 -2001. This year KaliW\\\ end at 12 PM on 4-1-2002 when people 
will be sleeping, So, kalixs called sleeping. 2 nd year dvaparaW\\\ end on 5-1-2003 at 6AM, so dvapara is called 
twilight. 3 rd year tretaW\\\ end at 12 AM on 5-1-2004, when people will be standing, so tretaxs called standing.4 th 
year will end on 4-1-2005 at 6PM as there is leap year in 2004. Here year is of 365 1 / days, (b) 5 year yuga - 
Yajusa jyotisa has described 5 years yuga in which lunar years match with solar year by adding 2 extra (adhika) 
months. Years are named by adding prefixes sam-, pari-, id-, ida-, anu-, to the word vatsara. (c) 12 years yuga 
is revolution period of Jupiter around sun. These are named like months of lunar year-chaitra, valsakha, etc. (d) 
Rahu yuga-\\ is called Saros cycle in Babylonian astronomy. This is relative motion of Sun and Rahu (node of 



Moon) in 18 years 10,5 days in which eclipse cycle repeats. Its half period of 3339 tithis is also approximately 
eclipse cycle stated by Visvamitra (Rk 3/9/9). (e) 19 year yuga — This is followed in Rk-jyotisa, as explained by 
Sri Prabhakara Ho/ay Nagpur. In this period, lunar years match with solar year more accurately (less than 2 
hours error) by adding 7 adhika months. Years are classified into 5 types according to 5 blocks of 6 tithis in 
which a solar year starts. In 1 yuga, there are 5 years of samvatsara type. 

(2) Manusya (human) yuga - 60 years active life of man is called Angira period in which 6o year cycle of guru-years 
occurs. In 100 years, Saptarsi (Ursa major, ursa -rsi) moves / naksatra, i.e.27 th part of zodiac circle. In Rk- 
jyotisa calculation, moon moves 1 naksatra ahead in 100 years. The line joining two eastern stars moves 1 nak- 
satra back in 100 years. This year count has been called Laukika in RajataranginT. 1/3 rd of divya-dina or pari- 
varta yuga of 360 years is 120 years which is human life for astrological timing of events. 

(3) Parivarta yuga-Th\s is divya-dina of 360 years in which historical changes (parivartana) occurs. 71 such yugas 
make manu-yuga of 26,000 years (precession period of earth's axis) in Brahmanda Purana (1/2/29/19) 

(4) Sahasra yuga-Bhagavata purana (1/1/4) states 1000 year satraof Saunaka in Naimisaranya. Compilation of Pu- 
rasas took about 200 years, but its effect on social norms lasted for thousand years. It could be revised only in 
time of Vikramaditya of Ujjain (82BC-19AD, era started in 57 BC) as per Bhavisya purana (3/3/1/2-4). During 
3000 years, seasons shifted back by 1 Vi months.720 years of MunTsvara is of 2 parivarta yugas. Sahasra 
(1080) years is 3 parivarta. Even Gautama Buddha planned his religion for 1000 years. Prophet Mohammed 
predicted Islam to last for 1400 years. Saptarsi yuga is of 2700 years. It is described in two ways in Brahmanda, 
Vayu puranas. 2700 solar years are called divya years, Manusa year is 12 revolutions of moon around earth in 
327.5364 days. Saptarsi eta is also stated to be of 3030 manusa years =2717 solar years. Romaka yuga of 
Pahchasiddhantika (Varahamihira) is of 2850 years =19 year Rk-yugax 150, 

(5) Dhruva or Krauhcha yuga-Jh\s is of 9090 manusa years or 8100 solar years. This is exactly 3 times saptarsi 
yuga and about 1/3 rd of Ayana or Manu-yuga. Position of north pole of earth makes a circle in 26000 years and 
is close to 3 stars so the period is divided into 3. On earth regions around north pole are called Krauhcha-dvTpa, 
so it is called Krauhcha yuga also. In north India, Guru years are calculated as per actual mean motion of guru 
in 361.14 days (Surya-siddhanta). In 85 solar years there are 86 guru years. In south India, solar years are 
named as guru years (Paitamaha-siddhanta). In 85 x 60 = 5100 years, both cycles are completed. On 11-2- 
4433 BC when Rama was born, the year was start of guru cycle in both systems (1 st Prabhava year) as per 
Visnudharmottara purana (82/7,8). Matsya incarnation had occurred 5100 years before that in 9533 BC. He- 
rodotus gives date of sinking of last island of Atlantis in 9564 BC. This is approximately period of last glacial 
flooding. 

(6) Ayana (Precession) yuga-Earth's axis rotates around pole of ecliptic (earth orbit) in 26, 000 years. This has been 

called Manu yuga in Brahmanda purana (1/2/29/1 9). Glacial age on earth is due to two cycles-Precession in 
26000 years in reverse direction and advance of earth aphelion in 1 lakh years. Glacial region is around north 
pole. When it is inclined away from sun or earth is at aphelion it gets less heat. When both combine, it is Glacial 
ice age. Its cycle is in 21600 years— 1/21600 = 1/26000 +1/100000 

Civil cycle is taken as middle of the two. By taking 24000 years cycle, there is positive error in 12000 years and 
negative in other half (Brahma-sphuta-siddhanta, madhyamadhikara, 60, Siddhanta-siromani of Bhaskara-ll, 
Bhu-paridhi, 7). Each half of 12000 years is taken as yuga of 12,000 divya years in every purana, Mahabharata, 
etc. In Mahabharata period AvasarpinF (descending) period was running in which part yugas -Satya, treta, dva- 
para, kali, of 4, 3, 2,1, parts come in that order. Of this order, /(a// started on1 7/1 8-2-31 02 BC Ujjain mid-night. 
The other half is called UtsarpinT (ascen<S\ng) in which kali to satya yugas come. 



(7) Astronomical yuga — This is 360 times ayana yuga of 43,20,000 years in which all planets up to Saturn complete 
integral number of revolutions (Bhaganopapatti'm Siddhanta-Siromani of Bhaskara-W). Two other cycles depend 
on it which is not verified so far-(i) Movement of magnetic pole and magnetic reversal, (ii) Movement of geo- 
graphical pole in north south direction (Indra-Vijayaoi Madhusudan Ojha) or equivalent continental-shift. 

Time has been equated to full-pot (Purna-kumbha) or volume .and parallel with 7 chhandas is shown in Kala-sukta of 
Atharva-veda (19/7), BrhatT-sahasra (36000) days is life period of man (Aitareya Brahmana). Understanding of these 
yugas explains puranic chronology since 62,000 BC. 

(Detail article with references is in Hindi, titled — " Chhanda-Adharita Mapa-Vijnana" 



Chapter - 2 
REVOLUTION OF PLANETS 

Subject - This chapter deals with total 
revolutions of planets in a kalpa, adhika masa (gain 
of lunar months above 12 in a solar year), ksaya 
tithi (shortage of lunar dates from months of 30 
civil days), rotation of orbits. From that; the average 
daily motion of planets have been calculated. 

1. Explanations - All the results given in this 
chapter are assumed and no hint is given as to 
how these numbers have been found. Obviously 
they are highly accurate and have been followed 
since time immemorial. All texts from 
Suryasiddhanta to Siddhanta Darpana have fol- 
lowed the same practice. 

It is possible that the samhita and Brahmana 
texts gave the methods and observations of 
planetary motions. The teaching of science and 
mathematics was like present day text books -of 
college, and not in verse form which is useful for 
memorising only. This became necessary when 
educational institutes and their books were 
destroyed due to foreign invasions. Mathematics 
in modern text book form has been found in 
Bakhsali manuscripts of mediaval preiod (edited in 
3 vols by G.R. Kay - New Delhi-7) 

All ancient authorities have admitted that 
these results are not based on observations. Surya 
siddhanta has stated that these were given by Surya 
to Mayasura in Romaka town in 21, 63, 223 B.C. 
(121 years before the end of Satya yuga). To some 
extent, it is correct. Even with most modern 



Revolution of Planets 47 

equipments, calculation of motion for billions of 
years cannot be made on observations during a 
life time only. It needs systematic observations for 
at least 500 years for studying motion in orbits, 
and at least 10,000 years and much more, if rotation 
of orbits, or change of earth's axis is to be calculated. 
Thus the result could have been otained only from 
observations through the ages, preserved by 
generations (like Vedas, it has to be 'Apauruseya' 
i.e. god given or beyond a human being). 

2. Origin of complete revolution numbers in 
a kalpa - There are two assumptions by ancient 
authorities - In general, it is assumed that the 
figures have been obtained on the basis of 
observations through ages. Total motion in a kalpa 
has been calculated on basis of observed rates. 

Siddhanta Darpana has followed pattern of 
Siddhanta Siromani of Bhaskara II except for some 
new improvements. Bhaskara has assumed that 
concept of yuga and kalpa has been derived from 
the observed motion of planets. The planets repeat 
their positions after every yuga (the grand year), 
as in civil year earth comes back to the same 
position round the sun. However, if we consider 
rotation of orbits; its cycle is repeated only after 
1000 yugas or a kalpa. For example, orbit of Saturn 
rotates only 39 times in a kalpa, so its motion 
cannot be perceived within a yuga. Text books of 
Tantra and karana are not concerned with such 
slow motion. 

Another presumption is that theories of 
planetary motion and constants of orbit have been 
given in vedas. We do not know the technical 
terms and method of presentation of astronomy as 
explained in samhita and Brahmana texts. 
Varahamihira was probably last who * understood 



48 



Siddhanta Darpana 



contents of all 3 parts of jyotisa from vedas. It is 
presumed that 10,000 verses of Rkveda contain 
records of astronomical observations for 10,000 
years or yugas of 5 or 19 years. Though only 
Aryabhata I has specifically mentioned two motions 
of -earth, it appears that many others knew about 
movement of earth. It is clear from names Jagat 
(moving), samsara etc. Methods of calculation 
followed by other astronomers also indicate that 
they were following some theories known in vedas 
or other texts but not specified in astronomy works. 
Whatever may be the nature of planetary motion, 
it contines to be observed against the background 
of same Zodiac of 12 rasis or 27 naksatras, and 
from earth only. From scattered observation charts 
through the ages, theories of circular or elliptical 
orbits have originated. They cannot be observed 
directly. 

3. Circle and Ellipse 





Circumference 

6 Centre, radius OP 

or OQ 

Fig. 1A-Circle 



AB = Major axis, CD = Minor axis 

O Centre, Focus F1 and F2 

Fig. IB-Ellipse 



Definition - Circle is path (locus) of a point 
(circumference) which remains at fixed distance 
(radius) from a fixed point called centre. (Figure 1 

A) 



Revolution of Planets 49 

Circle is a round figure on a paper looking 
same from all directions. Ellipse is elongated form 
of circle - stretched in two opposite directions 
(called major axis) 

Definition - Ellipse is locus of a point whose 
sum of distences from two points Fi and F2 (called 
focus) is constant. Thus DF1+DF2 = D'F 1 +D'F 2 = 
AB (major axis). Smallest width is minor axis CD. 
AB and CD are perpendicular at their middle point 
O, called centre (Figure IB) 

Kaplar's laws of planetary motion indicate that 
planets move in an ellipse round the Sun which 
remains at one of the focus (not at the centre). 
Newton's law of gravitation were derived from 
Keplar's laws and vice versa. 

In circular orbit (special type of ellipse where 
both focus are at same point), speed of planet will 
remain constant. In elliptical orbit, it will be fastest 
when the planet is closest to Sun (at A if sun is 
at Fi). It will be slowest when farthest from sun 
(at B). B is called Aphelion (Apex=top, helios = 
Sun) or mandocca (slow+top) or ucca in short in 
jyotisa. 

One feature of circular orbits remain the same 
in elliptical orbit al$o. Though the speeds vary at 
different position, the area covered by line from 
sun to the planet in unit time remains the same. 
Thus POQ area in circle or DFiD' area in ellipse, 
covered in unit time remain constant. 

4. Sighrocca and Mandocca - The relative 
motion of sun and earth remains the same, whether 
it is observed from sun or earth. In either case, it 
will be elliptical motion with same speed. Similarly, 



50 Siddhdnta Darpana 

moon also moves in elliptical orbit round the earth. 
The position of sun or moon where its speed is 
lowest (at highest point in orbit), is called man- 
docca. 

Orbits of other planets around sun are also 
elliptical. However, when we observe from earth, 
it is a composition of two elliptical motions - one 
is relative motion of sun around earth and the 
second is motion of planet round the sun. The 
planet in smaller orbit is called sighrocca, as the 
average motion is faster in smaller orbit. The 
highest point in slower and bigger elliptical orbit 
is called mandocca. 

In bigger orbit, a planef s motion will appear 
slow due to two reasons. At lafger distance, 
gravitational attraction of Sun is small and planet 
moves at small speed to counter the attraction by 
its centrifugal force. If speed is more, it will go 
still farther and loose speed, till it settles into a 
stable orbit. Due to larger distance the angular 
speed appears still slower. 

5. Pata- Orbit of sun around earth and orbit 



SS' = major axis of 
sun's orbit 
MM' = Major axis of 
Moon's orbit 
R, K = Point of inter- 
section of two orbits 




of moon around the earth are hot in the same 
plane. They are inclined at angle of about 5° (Figure 

2). 



Revolution of Planets 52 

If sun orbit is taken as reference level, at 
point R, moon appears moving towards north or 
upwards. Moon itself is not at R, it may be at any 
point on its orbit MKM'R. R is merely an imaginary 
point of intersection and is called uttara-pata 
(ascending mode) or Rahu. 

At point K motion of the moon appears south 
wards, hence it is called daksina-pata (descending 
node) or ketu. 

When moon is at one of the patas on purnima 
(180° away from sun) or amavasya (same direction 
as sun), eclipse occurs. Thus Rahu and Ketu are 
said to cause eclipse. Rahu and Ketu are called 
chaya graha as they are only imaginary points. 
They have nothing to do with shadow of earth or 
moon. R and K are always in opposite direction 
from earth as seen from the diagram. They are 
moving in reverse direction to the direction of 
planetary motion. Their revolution is called 
bhagana of pata (in about 19 years). 

Similarly, pata of other planets also move. But 
their motion is so slow that it is noticed only in 

a kalpa. 

Motion of Ucca - Motion of ucca of moon is 

visible in a yuga (one revolution in about 9 years). 

Motion of other planets is very slow and can be 

noticed only in a kalpa. 

6. Change in values of bhagana - Surya 

Siddhanta, first chapter states that motion of 

planets vary with time and hence its observation 

needs to be corrected after long lapse of time. 

It is known in modern astronomy that earth's 
rotation on its axis is slowing down at the rate of 



52 Siddhantd Darpana 

14 seconds per century due to tidal function. Due 
to decrease in angular momentum of earth, moon 
is moving away at the rate of 8mm every year to 
conserve the angular momentum of earth-moon 
system. Due to tidal forces of galaxy and sun and 
friction of solar atmosphere, motion of planets also 
will slow down. But its values are not known either 
in siddhanta texts or in modern astronomy. It can 
be inferred to some extent by comparison with old 
records of solar eclipse in vedas, or comparing old 
values of bhaganas with present values. 

Thus, if we calculate the average motion of 
the planets on the basis of their total motion, their 
values will differ from the real observation. Another 
reason of error will be inaccuracy and approxima- 
tion of mathematical methods and calculations. To 
correct these, various astronomers have introduced 
correction terms for their era. Candrasekhara was 
last among them. In addition, he introduced 3 
correction terms for moon's motion, whose error 
was noticed due to its faster motion. 

REFERENCES 

(1) For knowledge of circle and ellipse any college 
text book on plane coordinate geometry can 
be referred. For example Loney's coordinate 
geometry. 

(2) For concept of intersection of two orbital 
planes any book on solid geometry can be 
referred. Fuller discussion will be in books of 
spherical Trigonometry by Gorakha Prasada 
or by Todhunter. 



Revolution of Planets 53 

(3) Development of planetary theories of motion 
have been excellantly explained in 'The struc- 
ture of the Universe' by Sri J.V. Narllkara. 

(4) Historical discussion of zodiac and factional 
slowing of planetary orbits is given in 'An 
intelligent Man's guide to Science' by Isaac 
Asimov. 

(5) For comparison of values of bhaganas given 
in different texts any good commentary of 
standard texts may be referred. One may read 
the histories of astronomy, referred to earlier 
in introduction. 

Translation of the text 

Verses 1-2 - Bhaganas of planets in a kalpa 
(West to east) 

Ravi, Budha, Sukra 4,32,00,00,000 

Candra 57,75,33,36,000 

Mangala 2,29,68,71,112 

Brhaspati 36,41,55,205 

Sani 14,66,49,716 

Budha SIghrocca 17,93,69,67,141 

Sukra SIghrocca 7,02,22,57,860 

Mangala, Brhaspati, Sani SIghrocca 4,32,00,00,000 
Note : 1 - Sun - Ravi, Surya, Arka etc. in 

sanskrta 

Mercury - Budha; Venus - Sukra, Mars - 
Mangala, Kuja, Bhauma; Jupiter - Guru, 
Brhaspati; Saturn - Sani 



54 Siddhanta Darpana 

2. Budha, Sukra are in inner orbits around 
sun, so their revolutions are same as sun as they 
appear tied with it as seen from earth. Their 
revolution is equal to number of solar years in a 
kalpa by definition (1 year corresponds to 1 
revolution of sun) 

3. Sighrocca of Brhaspati, Sani and Mangala 
is due to earth's orbit round the sun. Hence it is 
equal to apparent revolution of sun round the 
earth. 

Verse 3 - Mandocca Bhagana in a kalpa from 
west to east. 

Siddhanta Darpana Surya Siddhanta 

Ravi 334 387 

Candra 48,81,17,940 48,82,03,000 

Mangala 310 204 

Budha 410 368 

Guru 805 900 

Sukra 557 535 

Sani 70 39 

Note - Only source of these figures is Surya 
siddhanta. Author has not indicated source of his 
corrections. 

Verse 4 - Bhagana of pata (East to West) 

Note - Pata is calculated according to inclina- 
tion of orbit with Ecliptic. Since it is path of sun, 
there is no pata for sun. 

Planets Bhagana in a kalpa Surya-siddhanta 
Candra 23,22,98,033 23,22,38,000 

Mangala 298 214 

Budha 552 488 

Guru 945 174 



Revolution of Planets 55 

Sukra 110 903 

Sani 545 662 

Note - Source of different figures and large 
variations in figures for guru and sukra is not 
explained. 

Verses 5-6 - Naksatra dina is the time between 
rising of any naksatra to its next rising (equal to 
time of revolution of earth on its own axis) 

The time between rising of a planet to its 
next rising is called savana dina for that planet. 
(For example sunrise to next sunrise is savana surya 
dina). This corresponds to rotation of earth with 
respect to that planet, 
(i) Total number of naksatra dina in a kalpa 

15,82,23,78,28,000 

(ii) Savana dina of a planet = Naksatra dina 

- graha bhagana 

(iii) Candra masa = Candra bhagana 

- Surya bhagana 

Verse 7 : In a kalpa (or a Mahayuga) 

No. of adhimasa = No. of candramasa - No. 
of Sauramasa 

No. of Kshaya dina (Lost days) = 30x No. of 
Candramasa - No. of savana days 

Verses 8-11 : 

No. of solar months in a kalpa 51 ,84, 00,00, 000 

No. of Candra months " 53,43,33,36,000 

No. of adhimasa " 1,59,33,36,000 

No. of Sauradina " 15,55,20,00,00,000 

No. of Candra dina " 16,03,00,00,80,000 



56 Siddhanta Darpana 

No, of Surya (savana) dina " 15,77,91,78,28,000 

No. of Ksaya tithi 25,08,22,52,000 

Note - No. of savana dina and ksaya tithi 
here is same as that of Surya siddhanta, where 
the figures given are for a mahayuga. However, 
savana days in a mahayuga are different according 
to other texts - 

Surya siddhanta of Panca siddhantika - 1,57,70,17,800 

Aryabhata - 1,57,79,17,500 

Brahma sphuta siddhanta, siddhanta 

Siromani ' 1,57,79,16,450 

Mahasiddhanta - 1,57,79,17,542 

Verse 12 - Definition - At any given time 
kendra of a graha (angle) = position of a planet- 
position of its ucca. 

Compared to Sighra ucca it is called SIghra 
kendra, compared to manda ucca, it is called manda 
kendra. 

Ucca and pata bhagaras are not completed in 
a yuga except for moon, so their bhaganas are 
stated for a kalpa (1,000 yugas) 

Verse 13 : Sighrocca = drak, cala, asu, capala 
etc. 

Mandocca = Mrdu, ucca, manda etc. 
(synonyms) 

Verse 14-15 - No. of asu (prana = 4 seconds) 
in a day- 

1 average (madhyama) naksatradina = 21,600 



asu 



1 madhyam saura dina = 21,976 asu 
1 madhyam Candra dina = 21,320 asu 
1 madhyam savana dina = 21,659 asu 



Revolution of Planets 57 

Savana dina is commonly used by people 
which is divided into 60 ghatika or danda. 

Verse 16 - Bhagana = 1 complete revolution 
= 360° amsa. Bhagana kala = bhagana x 360 x 60 

Dainika kala of a graha = graha bhagana kala 
in kalpa/savana dina in kalpa 

Time for 1 bhagana of a graha = savana days 
in kalpa/graha bhagana in kalpa 

Verses 17-18 : Like division of full circle rota- 
tion in 360° (amsa) and then further sub-divisions 
by 60 in each step, learned men have divided a 
savana dina also by 60 at each step to danda and 
pala etc. One complete revolution (bhagana) of sun 
takes days 365/15/31/31/24 danda, pala etc. 

Verse 19 - Madhyama guru takes days 
361/5/27/27/13 in one rasi at average speed. 

Verses 20-24 - Daily motion of planets is 
described in liptas (1/60 amsa) and 10 furhter 
sub-divisions in steps of 60. By multiplying this 
daily motion with no. of days (passed from 
beginning of kalpa to desired day), madhyama 
graha (position with average speed) is obtained. 

Sun (Ravi) 59-8-10-10-24-12-30-4-10-4 

Candra , 790-34-52-3-49-8-2-16-10-11 

Mangala 31-26-30-6-47-44-32-49-3-4 

Budha SIghra 243-32-16-7-17-17-59-43-42-44 

Guru 4-59-5-37-0-36-41-17-1-51 

. Sukra SIghra 96-7-37-47-57-50-39-32-31-35 

Sard 2-0-26-55-2-53-21-2-4-54 

Candra ucca 6-40-54-31-0-44-5-52-45-39 

Candra pata 3-10-47-40-40-26-11-25-13-30 



58 Siddhanta Darpana 

Verse 25-26 - Kranti vrtta in sky is the 
sudarsana cakra of Jagannatha with which he 
removes fear, produces light and destroys all in 
the end. With this prayer Sri Candrasekhara Simha 
completes second chapter of Siddhanta Darpana 
describing bhagana of grahas. 




Chapter - 3 

MEAN PLANETS 

Scope - This chapter describes methods for 
calculating value of madhya graha (position calcu- 
lated from average motion). This coincides with 
sphutagraha (true position) twice in every bhagana 
(revolution). Since the planetary motions started 
from mandocca position; at mandocca, sphuta and 
madhyama positions should be same (for sun and 
moon). 

Verse 1 - Ahargana (count of days) - Ahargana 
for ista dina (desired day) is counting of days from 
beginning of kalpa (in siddhanta text). This is 
needed to know the graha on ista dina of any 
varsa, masa or tithi. 

Note - In tantra, ahargana is counted from 
beginning of mahayuga (or ^sometimes, from the 
beginning of kaliyuga). In karana text, ahargana is 
counted from any reference year or beginning of 
current year itself for preparation of panjika. 

Verses 2-8 - Steps in calculation of ahargana - 

1. Add the saura varsas for 6 manu, 7 manu 
sandhi (each equal to satya yuga), 27 mahayuga, 
3 padayuga and years passed in current kaliyuga. 

Note : In the present Svetavaraha kalpa, 6 
manus out of 14 have passed. In the current 7th 
vaivasvata manu, 27 yugas have passed. At 
beginning of kalpa and after each manu, a sandhi 
equal to one satya yuga exists! In current 



60 Siddhanta Darpana 

mahayuga, satya, Treta and Dwapara have passed. 
Kaliyuga started on 17/18-2-3102 B.C. Ujjain mid- 
night. 

2. Deduct 1, 70, 60, 400 years 

Note - According to verse 24 of 
madhyamadhikara in Surya Siddhanta, Brahma 
took this time of 47,400 divya varsa to create stars, 
planets and living beings. The present stable 
motion of planets started after that. 

3. Multiply by 12 to make it months and add 
the number of months (masa) elapsed from Caitra 
(Candra months in current year are almost equal 
to saura masa). 

4. Keep the result (no. of completed saura 
months) at two places. 

5. At first place, multiply it by no. of adhika 
masa (1,59,33,36,000) in a kalpa and then divide it 
by sauramasa in a kalpa. Result will be adhimasa 
related to the saura varsa. 

6. Add this to the no. of masa from kalpa 
beginning obtained at step 3. 

7. Multiply Candramasa by 30 and add the 
days completed in the present month (Candra 
masa) 

8. Keep the result at two places. 

9. At one place, multiply it by kalpa tithi 
ksaya (25, 08, 22, 52, 000) and divide by number 
of kalpa tithi. Substract the result from kalpa tithi 
at the second place. Difference is number of savana 
tithis from kalpa beginning. Divide it by 7. 
Remainder will give the week day counted from 
ravivara (sunday) as 1. 



Mean Planets 61 

Mathematical comments - 1. The methods are 
based on rule of 3 (Trairasika) or ratio and 
proportion. 

, v Adhimasa till ista dina Adhimasa in a kalpa 

/^\ U _ k. 

Sauramasa till ista dina Sauramasa in a kalpa 
... ksaya tithi till ista tithi ksaya tithi in a kalpa 
^ ' gata tithi (elapsed tithi) Total tithi in a kalpa 

Tithi is a candra dina. 

2. Ratio between Candra and saura masa, 
tithis; Saura masa + adhimasa = Candra masa 

Within current year, they are almost equal. 

Candra masa x 30 = Candra tithi 

Candra tithi is almost equal to savana dina 
in a current month 

Candra tithi - ksaya tithi = savana dina. 
Savana dina is time from sunrise till next sun rise. 

3. Kalpa had started on ravivara at midnight 
at Lanka which is at equator on 0° longitude of 
India (passing through Ujjain). 

Verses 9-13 : Errors in approximation of 
sauramasa and Candra tithis (as explained in 
mathematical notes above si 2) 

Adhimasa - While calculating adhimasa only 
the quotient (result) is taken and remainder is left 
out. If remainder is almost equal to divisor, or if 
an adhimasa has passed recently (in past 1 year), 
then 1 is added to know correct adhimasa. 
However, if the remainder is almost zero or an 
adhimasa is to come soon, then 1 is to be 
substracted. 

Ksaya tithi - Similarly, if in calculation of 
ksaya tithi, remainder is almost equal to divisor 



62 Siddhanta Darpana 

and within a week ksaya tithi has passed, then 1 
is added to the result. (If pancami comes after 
tritiya, then caturthi is ksaya tithi upto dasaml, 
if remainder is more than half the divisor, 1 is to 
be added to ksaya tithi. Thus 1 will be substracted 
from ahargana. If remainder is almost zero, 1 is 
added to ahargana. Correctness of ahargana can 
be checked with week day. 

Verse 14 - Masadhipati - Divide ahargana by 
30, multiply the result by 2, add 1 and divide by 
7. Remainder will indicate week days counted from 
ravivara as 1. (soma 2, mangala 3, budha 4, guru 
5, Sukra 6, Sani 0) Ruler of this day will be 
masadhipati. 

Derivation - Each civil month is of 30 days 
(civil). Ruler of 1st day is masadhipati. Ahargana 
divided by 30 gives the number of civil months. 
In each month of 30 days; 4 weeks are completed 
(4x7=28 days) and 2 days remain. Hence for each 
month; 2 remainder days are taken. 1 is added 
because the first day of kalpa was ravivara, 1st 
day. 

Verse 15 - Divide ahargana by 30. Remainder 
is the days gone (gata dina) in current month. 
Gata dina substracted from 30 gives bhogya 
(remaining days) dina of the month. 

Derivation is obvious from earlier verse. 
Verse 16 * Divide ahargana by 360, Multiply 
result by 3 and add 1. Divide the result by 7. 
Remainder indicates week days starting from ravi 
as 1, which is the varsapati. The remainder left 
after division of ahargana is bhukta dina (past days) 
of current year. 



Mean Planets 63 

Comments : (1) Masasidhipati and 
varsadhipati are used only for calculating kala bala 
in horoscopes, or in mundane astrology for 
forcasting events of the year. It has no importance 
in ganita jyotisa. 

(2) Each civil year is of 360 civil days. Hence 
the quotient after division by 360 into ahargana, 
is number of completed civil years. Remainder will 
be past days of the current year. 

(3) In 1 year of 360 days, 360 -s- 7 = 51 weeks 
and 3 extra days remain. Hence each completed 
year gives 3 days for count of week days, Next 
day will be first day of current year, hence 1 is 
added to find varsadhipati. 

Verse 17-20 - Lord of first day of masa (month) 
is masadhipati, and lord of first day of varsa is 
varsadhipati. 

Satananda (author of Bhasvati karana) and his 
followers have different opinion. Lord of the day 
on which mesa samkranti falls is the lord of the 
year (varsadhipati). To calculate the number of days 
in that year, the following rule has been given. 

Calculate the danda, pala etc. from time of 
entry of ravi in mesa samkranti to the time of 
beginning of next day. Multiply it by 4 and keep 
it in 3 places. Divide the number at third place by 
37 and add the result at second place. Divide at 
second place by 8 and add this result at first place. 
The result in danda etc. will indicate the number 
of days for which the varsapati will rule. For 
remaining days of the year, (360 - days of rule of 
varsapati) lord of day next to sankranM will rule. 



Siddhanta Darpana 
64 

According to this rule, no graha can rule for more 
than 271 days. 

Mathematical symbol : let T = time in danda 
etc. from entry of ravi in mesa to next sunrise. 



4T 

37 



= T'+R (remainder smaller than 37) 



4T + T _ j " + R' (remainder smaller than 8) 

8 
4T+T" = D danda + p pala etc. 
D is the number of days for which varsapati 
will rule. Lord of the day after mesa Sankranti will 
rule for 360 - D days. 

2 This appears to be a convention by 
Satananda, hence no derivation of the rule is given. 
3. Maximum days of rule of varsapati - 
T < 60 danda 

4T 4X60 = fi5 
T < 37 Hi7~ 

4j + j' 4 X 60 4- 6.5 t 

t» < — g- < r^ 

D = 4T + T" < 4 x 60 + 30 = 271 
Hence maximum days of rule from sankranti 

day is 271 days. 

Verses 21-22 - Formula for calculating graha 
for indicated day - Multiply ahargana by kalpa 
bhagana and divide by kalpa savana du£ ResuU 
wiU be lapsed bhagana. Multiply remainder by 12 
and again divide by kalpa savana dina. Again 
multiply by 30, 60 and 60 and divide by kalpa 



Mean Planets 



65 



savana dina to obtain amsa (degree) kala (minutes),, 
vikala (seconds) 

Explaination (1) By ratio and proportion 

Bhagana till ista dina _ Ahargana 

Bhagana in a kalpa ~ Savana dina in a kalpa 

(2) Fraction of bhagana are converted to rasi 
etc. according to the scale - 

1 Bhagana = 12 rasi, 1 rasi = 30 amsa 

1 amsa = 60 kala, 1 Kala = 60 Vikala 

+ 

(3) 1st rasi is mesa starting from 0° to 30° in 
kranti vrtta (ecliptic). 0° starts from a fixed point 
marked by star groups in Indian astronomy. In 
western system, 0° is marked by point of intersec- 
tion of equator with ecliptic plane, where- motion 
of sun appears northwards. Difference between the 
two initial points is called ayanamsa. Axis of earth 
rotates one round in 25,762 years. In Indian system 
also calculation of day length, lagna etc. are done 
from this ayanamsa sayana point. 

(4) 12 rasis are 1. mesa 2. vr$a, 3. mithuna, 
4. karka, 5. simha, 6 kanya, 7. Kite, 8. vrscika, 9, 
dhanu, 10. makara 11. kumbha, and 14. mina. 

Verse 23 - (Quoted from Surya Siddhanta) - 
Same method is used for calculation of Slghrocca, 
mandocca and pata for ista dina. However, for 
pata, the result will be deducted from 12 rasi, 
because movement of pata is in opposite direction 
of graha. 

Note - When it is unnecessary to explain in 
more detail, the author has just referred to 
quotation from previous authorities - mainly surya 
sidhanta or siddhanta siromani. Sometimes quota- 



66 Siddhdnta Darpana 

tions have been given for comparison or contradic- 
tion on important points. 

Verses 24-25 - Calculation of guru varsa - 

calculate bhagana of guru as before and add 3 
(bhaganas) Multiply the sum by 12 and add their 
rasis lapsed and add 2 again. Divide this sum by 
60 and add 1 to the remainder which indicates 
guru varsa counted from Prabhava etc. 

Notes: (1) Secret of guru varsa has been 
explained in chapter 21 of this book. 

(2) Guru takes about 12 years to move around 
sun and about 1 year to cover 1 rasi. Hence guru 
varsa (time in a rasi with medium speed) is similar 
to saura varsa (time of 12 rasis or complete 
bhagana) Guru varsa is called samvatsara of 
361.02672 savana days which is smaller by 4.23203 
days from saura varsa and bigger by 1.02672 days 
from savana varsa of 360 days. 

(3) 60 years are needed to complete 5 
revolutions of guru and 2 revolutions of Sani. Thus 
a cycle of 60 years has been adopted for samvatsara 
of guru. This is the active life period of a man. 

(4) Guru varsa are listed in verses 32-46. 
Varahamihira in Vrhatsamhita has assumed the 
beginning of samvatsararakra from 35th samvatsara 
Prabhava, instead of the first vijaya. However, the 
calculation method given here will start guru, 
samvatsava from the 13th 'vikrama', for start of 
first rasi. Thus one complete round of 12 rasis in 
12 samvatsaras is considered complete at beginning 
of guru motion. This is only a convention. Same 
result could have been obtained by calculating rasi 



lAean Planets & 

f madhyama guru and count the samvatsara from 

13th. 

(5) Symbolic formula 

(a) Madhya guru = B bhagana + R rasi + A 
amsa etc. 

(b) Total samvatsara = (B+3)xl2+R+2 = S 

(c) S/60 - s+r (remainder to 59) 

(d) r+1 is 1 to 60 samvatsara counted from 
prabhava. 

(6) Samvatsara for 1st rasi-completed R=0 / B=0 
n = (r+1) counted from 35th samvatsara 

x « S (B + 3) x 12 + R + 3 

12 B + R + 38 + 35 38 + 35 
- = = 13 remainder 

60 60 

Verse 26 - Elapsed part of guru varsa - (Omit 
bhagana and rasi from madhyama guru). Multiply 
amsa by 12 and add its 1/330 part which indicates 
elapsed days of samvatsara. (gata dina). (Deduct 
it from 361.02672 to find remaining days i.e. bhogya 
dina) 

Explanation - 30° of rasi - 361.027 days 

361.027 _ 1.027 



r = -^-= 12 + -35- days 



1° 12 x 



1 + 



1.027^ _ T 1 ^ 

= 12 x 11+ 



360 



330 



approx. 



Verse 27 - If in a Candra varsa, madhyama 
guru does not move to different rasi, it is called 
adhivatsara. (Guru varsa is 7 days bigger then 
Candra varsa and it may not complete 1 raSi in 
that period. 



68 Siddhanta Darpana 

Verse 28 : If with sphuta gati guru crosses 
two rasis in a saura varta, then it is called lupta 
varsa (samvatsara) (Normally guru will touch 2 rasis 
every saura varsa which is only 4 days bigger) 
unless both years start almost at sometime within 
4 days gap. However, if its true motion is faster, 
and years start almost same time, it may touch the 
third rasi also at end of saura varrha) 

Verse 29 - If in a saura varsa, guru in its 
sphuta motion goes to next rasi at higher speed 
(aticara), and does not return to the same rasi, 
that year is called mahacara kala. This year is as 
bad and inauspicious as a lupta samvatsara. (In 
this year also sphuta motion is faster than 
madhyama gati, not compensated by reverse 
motion. But guru may not cross into 3rd rasi, if 
its samvatsara does not start with saura varsa). 

Verses 30-31 : 60 Barhaspatya varsa contain 
12 Barhaspatya yuga (of 5 years each). 

Divide current number of barhaspatya years 
by 5, add 1 to the result. Sum is guru yuga starting 
from Acyuta etc. Within the yuga, the years are 
named according to remainder as 'sam', pari, ida, 
'anu' and 'id' vatsaras. Their adhipatis are agni, 
surya, candra, brahma and Siva respectively. 

Comments : This classification of vatsaras was" 
done in vedanga jyotirha. In one yuga of 19 years, 
there were five types of years. The years starting 
from 1st to 6th lunar tithi was called samvatsara. 
Years starting (solar) from next block of 6 candra 
tithis were called pari, ida, anu and id vatsaras 
respectively. In a yuga of 19 years, there were 5 
years of samvatsara type. Subsequently in yajur 



Mean Planets 



69 



jyotisa, a yuga was of 5 years, each of the 5 vatsaras 
occuring once. Same names have been adopted for 
barhaspatya yugas also. 

Verses 32-46 : Names of barhaspatya yugas, 
varsa and good or bad years - 

Yuga (adhipatis) years Subha(s) or Asubha (A) 

1. Visnu 
(Visnu) 





1. 


Prabhava 






2. 


Vibhava 






3. 


Sukla 


all Subha 




4. 


Pramada 




t 


5. 


Prajapati 




2. Barhaspatya 


6. 


Angira 


S 


(Brhaspati) 


7. 


Srimukha 


S 


(First yuga 


8. 


Bhanu 


A 


according 


9. 


Yuva 


S 


to our method 


10. 


Dhata 


A 


of calculation) 








3. Sakra 


11. 


Isvara 


S 


(Sakra) 


12. 


Bahudhanya 


S 




13. 


Pramada 


A 




14. 


Vikrama 


A 


i 


15. 


Vrsa 

* * 


S 


(Guru will cross vrsa rasi, when vrsa 


samvatsara will 


start). 








4. Pavakiya 


16. 


Citrabhanu 


. 


(vahni) 


17. 


Subhanu 






18. 


Tarana 


all asubha 




19. 


Parthiva 






20. 


Vyaya 




5. Tvastra 

* * 


21. 


Sarvajit 


S 


(Tvasta) 


22. 


Sarvadhari 


S 



70 



Siddhanta Darpana 







23. 


Virodhl 


A 






24. 


Vikrti 


A 






25. 


Khara 


A 


6. 


Ahirbudhnya 


26. 


Nandana 


S 




(Ahirbudhnya) 


27. 


Vijaya 


S 






28. 


Jaya 


S 






29. 


Manmatha 


A 






30. 


Durmukha 


A 


7. 


Paitrka 

• 


31. 


HemalambI 

• 


S 




(Pitara) 


32. 


VilambI 


S 






33. 


Vikari 


A 






34. 


Sarvari 


A 






35. 


Plava 


A 


8. 


Vaisva 


36. 


Sokakrta 

• 


S 




(Visvedeva) 


37. 


Subhakrta 

* 


S 






38. 


Krodhi 


A 






39. 


Visvavasu 


A 






40. 


Paravasu 


A 


9. 


Candra 


41. 


Plavanga 


A 




(Nisapati) 


42. 


Kilaka 


A 






43. 


Saumya 


S 






44. 


Sadharana 

* 


S 






45. 


Virodha krta 

* 


A 


10 


. Aindranala 


46. 


Paridhavi 


S 




(Indra and 


47. 


Pramathi 


S 




Agni) 


48. 


Ananda 


S 






49. 


Raksasa 


A 






50. 


Anala 


A 


11 


. Asvina 


51. 


Kapila 


A 




(Asvini 


52. 


Kaia 


A 




kumara) 


53. 


Siddhartha 


S 






54. 


Raudra 


A 






55. 


Durmati 


A 


12 


. Bhagya 


56. 


Dundubhi 


A 




(Bhaga) 


57. 


Rudhirodgari 


A 






58. 


Raktaksa 


A 



Mean Planets n 

59. Krodhana A 

60. K£aya A 

Verse 47 - Surya and Candra complete their 
full bhaganas in a mahayuga or in a padayuga. 
Hence their madhyamana can be calculated even 
from ahargana for mahayuga or for any padayuga 

also. 

Verse 48 - Another short method of finding 
ahargana is described below. It is not a fault for 
being a repetition, as great poets like Sri Harsa 
also have adopted such practice. 

Verse 49 : Multiply years since beginning of 
creation by 12 and add completed months from 
caitra sukla pratipada. Keep it in two places. At 
one place multiply it by 1,00,00,000 and divide by 
32,53,55,104. Add the quotient to result in second 
place. Multiply the result by 30 and add complete 
days passed after amavasya. Keep it again at two 
places. At one place multiply it by 1,00,00,00,000 
and divide by 63,90,97,35,058. Deduct quotient 
from quantity in second place. Result will be 
ahargana from beginning of creation counted from 
midnight of Lanka. 

Derivation of Formula 

Saura varsa x 12 = saura masa 

Completed Candra masa from caitra pratipada 
is assumed equal to saura masa. This approximation 
does not affect the result as the remainders found 
in calculation of adhimasa or ksayatithi are not 
used. 

Total saura masa x 30 = saura dina. 
Candra tithi after amavasya are # similarly 
assumed equal to saura dina. 



72 Siddhanta Darpana 

No of adhimasa 

- T r # - Adhimasa in a kalpa 

= No. of sauramasa (s) x 



Saura masa in a kalpa 
1,59,33,36,000 _ 1,00,00,000 

51,84,00,00,000 X 32,53,55,104 a PP rox ' 

This is added to sauramasa to get candra 
masa. 

candra masa x 30 = candra tithi 

Ksaya tithis till ista day 

__ No. of sauradina Ksaya tithi in kalpa 

till ista day (D) Sauradina in kalpa 

_ 25,08,22,52,000 _ 1,00,00,00,000 

15,55,20,00,00,000 " D x 63,90,97,35,058 ap " 
prox. 

We keep the significant digits same, so the 
approximation is sufficient for knowing integral 
numbers of adhimasa or ksaya tithi. 

Verse 50 : For calculating aharganas from kali 
beginning, the same procedure will be followed. 
However, 4 zeros from the multipliers will be 
removed and 4 last digits of divisions (5104 and 
5058) also will be taken out. Kaliyuga started on 
sukravara; so days will be counted from friday. 

Note : Kaliyuga = 1/10 yuga 1/10,000 kalpa. 
Hence 4 less no. of digit are required for 
approximation. Thus multipliers and divisors each 
are divided by 10,000. 

Verse 51 - Kalpa bhagana is multiplied by 
1811 and divided by 4000 to get bhagana at the 
end of dvapara. If the madhyama graha calculated 
from kaliyuga first day to ista day is added, madhya 
graha from beginning of kalpa is obtained. 

Derivation : Total yugas in a kalpa = 1,000 



Mean Planets 73 

Total yugas upto dvapara end 

6 manus x 71 = 426 yuga 

7x4 14 

7 sandhya x satyayuga = = — yuga 

4+3+2 9 

Satya + Treta + dvapara = — = — yuga 

79 
Time in creation = — yuga (to be deducted) 

Hence total yuga upto dvapara end is 

14 9 79 1 1811 

426 + 27 + _ + ___ = 453 __ = _ 

Bhagana at dvapara end _ Yuga at dvapara end 1811 
Kalpa bhagana Yuga in a a kalpa 4000 

Verses 52-55 : Position of graha, at kali 
beginning (midnight of 17/18 February 3102 B.C. 
at Lanka) are given below in vilipta (seconds). 

12,41,568 Candra mandocca 4,34,160 

mangala mandocca 4,56,840 

Budha mandocca 8,13,240 

gum mandocca 6,01,020 

Sukra mandocca 2,35,548 

Sani mandocca 8,97,480 

guru pata 2,55,960 

Sukra pata 1,96,020 

Sani pata 3,25,620 

At the time of writing Siddhanta Darpana, 
kali year 4970 end has been taken as reference year 
(karanabda). Deduct this number from the number 
of years passed since kali. Add 12 zeros to the 
right and divide by 2,73,77,85,151. The result will 
be gata dina from somavara. Ahargana will be from 
end day of sphuta mesa sankranti (year 1869 A.D.). 



Mangala 


12,41,561 


Budha sighra 


1,13,724 


guru 


82,620 


Sukra sighra 


1,49,040 


Sani 


11,91,02 


Surya mandocca 


2,83,176 


Candra pata 


7,14,788 


Mangala pata 


1,04,328 


Budha pata 


1,06,271 



74 Siddhanta Darpana 

Deduction : This is calculation of savana dina 
in a solar year. 

In a kalpa of 4,32,00,00,000 solar years, no. 
of savana dina is 15,77,91,78,28,000. 

So, savana dina in ista year (D) 

15,77,91,78,28,000 " , 
= 4,32,00,00,000 X ™ of y* ars M 



or D = y x 



15,77,91,78,28 10 12 



4,32,00,00 2,73,77,85,151 

First day of karanabda was monday. This will 
be ahargana till completion of year on mesa 
sankranti of madhyama surya. 

Verse 56 - Normally madhyama surya enters 
mesa, 3 days after entry of sphuta surya. So this 
third day after sphuta mesa sankramana, 1 
ahargana or main day of pancanga is taken. 
Therefore, madhyama graha is to be calculated for 
previous day of madhya mesa sankranti or on 2nd 
day of entry of sphuta surya in mesa. Then 
difference of grahagati for 1 day is to be added 
for madhyama graha of ista dina. 

-Verse 57 - There are different practices in 
different countries. Some pancangas take the entry 
of sphuta surya in mesa. Many pancangas take 
caitra sukla pratipada as 1st day. After madhyama 
saura varsa end, karanabda (4970 kali or 12-4- 1869 
A.D.) started. Author has given madhyamanas of 
dhruva (rasi at the beginning of year), ucca, pata 
etc. That day was soma vara (monday) and spasta 
surya had just entered mesa at sunrise. 

Verse 58 - Now madhyama dhruva (mean 
constants) for graha, mandocca, slghrocca, pata etc. 
are stated for somavara day before karanabda at 



Mean Planets 75 

time of sunrise at lanka (0° meridian through ujjain 

at equator) 

Verse 59-69 - Table of Karanabda dhruva - 
(in rasi / amsa / kala / vikala / para) 
(For 12-4-1869, Lanka sun rise) 



Ravi 

Candra 

Mangala 

Budha Sighrocca 

Guru 

Sukra Sighrocca 

Sani 

Ravi mandocca 

Candra mandocca 

Mangala mandocca 

Budha mandocca 

Guru mandocca 

Sukra mandocca 

Sani mandocca 



11/28/15/20/46 

0/3/20/29/53 

5/1/24/17/25 

10/18/14/9/2 

0/3/45/1/21 

11/13/41/42/12 

7/18/12/17/24 

2/18/47/54/0 

10/22/34/59/4 

4/7/1/42/13 

7/16/4/10/16 

5/17/17/0/15 

2/5/39/38/29 

8/9/19/44/10 



Pata dhruva of candra are corrected for reverse 
movement (bhacakra Suddhi is substraction from 
12 rasis) 



Candra pata (Rami) 
Mangala pata 
Budha pata 
Guru pata 
Sukra pata 
Sani pata 
Ketu pata 



3/21/19/18/28 

0/28/51/23/4 

0/29/17/28/58 

2/11/3/15/59 

1/24/3/31/0 

3/0/13/27/24 

9/21/19/18/28 



Verse 70 : The dhruva above have been 
calculated according to proportion ^of kalpa 



76 Siddhdnta Darpana 

bhagana. Candra pata is called Rahu, 6 rasi or 180° 
away from that is ketu pata. 

Verse 71 : Method to calculate mandocca and 
pata for past days has already been described. 
Mandocca and pata for a particular year can be 
calculated by this method. Multiply ista varsa by 
kalpa bhagana and divide by 2,00,000 which will 
tell the position in lipta etc. 

_ . t . Ista varsa Ista bhagana 

Derivation - —7 = — «-* — 

Kalpa varsa Kalpa bhagana 

or Ista bhagana - Ista varsa x — -^ ^- 1 - 

' * • ' Kalpa varsa 

Kalpa bhagana x 360 X 60 lipta 

= Ista varsa x - 2 — : *- — 

4,32,00,00,000 

Kalpa bhagana 

= 1?{a var?a x 2,00,000 ' u P a 

Verse 72 : Add this result to karanabda dhruva 
(deduct from pata) to get ista graha, ucca, pata 
etc. Alternatively, this can be calculated from 
annual motion (hara) also. 

Verse 73-74 - Hara (annual motion) in lipta 
is obtained by dividing kalpa bhagara by 2,00,000. 
Multiply elapsed years after karanabda (gata varsa) 
and add to dhruva to get ucca, graha etc. 

Verse 75 - Table of pata hara - 

Ravi mandocca hara 599 Guru mandocca hara 248 

mangala mandocca hara 645 Sukra mandocca hara 359 

Budha mandocca hara 488 Sard mandocca hara 2857 

Mangala pata hara 671 Budha pata hara 362 

Guru pata hara 1818 Sukra pata hara 212 
Sani pata hara 367 

Verse 76 - (Normally all astronomers assume 
that mandocca and Sighrocca move from west to 



Mean Planets 77 

east). Author says mandocca of mangala, budha 
and sani and slghrocca of Budha moves in both 
directions. This will be discussed while calculating 
true motion (graha sphuta) 

Verses 77-78 : While praying to lord 
Jagannatha in end, author states position of nilacala 
(Purl temple). It is 284 yojana north of equator 
on sea coast and 184 yojana east from 
Indian 0° longitude (Ujjain). 




Chapter - 4 
CALCULATION AT DIFFERENT PLACES 

Scope - In chapter 3, madhya graha etc were 
calculated for Lanka. In this chapter, calculations 
will be done for any place on earth. 

Mathematical Notes and definitions - 

r 

(1) Trigonometrical ratios- 

L ACB = 0, /.ABC is a right angle 

Then the , following ratios 
depend only on the value of 
angle 0, and not on the lengths 
of the sides of triangle. By 
definition these ratios are - 

Fig. 1 




SinG = 



Cos0 = 



Tan0 = 



AB 
AC 

BC 
AC 

AB 
BC 



Cote = 



Sece = 



tane 
1 

cose 



Cosec e = 



1 



sin e 




Fig. 2 



Calculation at Different Places 79 

(2) Indian Terms - To avoid decimals, a circle 
of circumferance 21,600 units, i.e. radius of 3438 
units is taken. One unit of circumference is equal 
to 1 kala, then 21,600 kala = 360° = 1 revolution. 

We draw OA and OB, two radii 

such that Z.AOB = 6 

Jya of LB is AC = R sin 

or sin x 3438 kala 

AC is half of the chord AD which is like 
string of bow shaped arc ABD. Hence its name is 
Jyarddha or Jya in short. 

OC is kotijya =R Cos = 3438 x Cos 

Tangent on A, meets base OB at E. , 

AE/OA = tan or AE = OA. tan = R tan 


Hence this ratio is called tangent or tan in 
short. In sanskrta it is called sparsa jya. OE pierces 
like arrow, hence called chedjya. OE = OA sec 
= R sec (sec is short of secant), Complement of 
angle 6 i.e. 90°- is called koti of the angle. Thus 
koti jya = jya of koti, 

koti sparsa jya = sparsa of koti 

and Koti chedajya = chedajya of koti 

In sanskrt another ratio is defined, called 
utkrama jya which is CB = R (1-cos 0). 

(3) Ratio of circumference to diameter is fixed 
and is called it (a greek letter, pronounced as 'pai') 
in modern mathematics. It is a transcendental 
number which cannot be expressed by any exact 
number. It can be expressed as non-recurring 
non-terminating decimal number to any desired 



80 Siddhanta Darpana 

approximation. Values upto 1,00,000 decimal places 
have been published. Calculation was on computer 
by the formula 

-i 1 -1 1 -1 1 

7i = 24 tan - + 8 tan 1 — + 4 tan A — 

8 57 238 

tan" 1 A is an angle such that 6 = tan A. It 
can be expressed as an infinite convergant series 
when A is smaller than 1. 

22/7 and 355/113 are rough practical ap- 
proximations of it correct upto 2 and 6 places of 
decimal respectively. If paridhi is expressed in kala, 
radius is 3437 3/4 kala approximately, which is same 
as 1 radian angle. (1 radian is an angle made by 
arc equal to radius) 

Madhava of Sangamagrama (kerala) in 13th 
century used infinite series to calculate value of n 
up to 30 places and sine table upto 9 places. Value 
of n up to 30 places have been expressed in a 
verse by him (read with katapayadi notation) - 

Accordingly, 

circumference _ 
diameter 

3.14,15,92,65,35,89,75,43,23,84,52,64,33,83,279 — 

(4) Yojana - Yojana is a measure of length as 
explained in the first chapter. Siddhanta darpana 
takes yojana of 1600 hasta = 24,000 feet or 7.3152 
kms approx. (if 1 hasta is taken as 18"). It takes 
diameter of earth as 1600 yojana then it is about 
4.94 miles approximately (hand will be about 19.6"). 



Calculation at Different Places SI 

(5) Longtrude, Latitude and sphuta paridhi - 

Study of sides and angles on a sphere is subject 
of spherical Trigonometry. It is called gola pada in 
jyotisa. 

To know position of a point in space by 
measuring its angle or distance from fixed point 
and lines is the basis of coordinate geometry (or 
cartesian geometry in the name of Rene de-Cartes 
of France, the originator). In a plane, two systems 
are used to indicate location of a point. 

P 







y ..p 




y 


;y 


X' 





X 

Y 




X 

Cartesian Co-ordinates Polar Co-ordinates 

Fig. 3a Fig. 3b 

In both systems, O is the fixed point called 
origin and a line through it OX is called X axis. 
In cartesian coordinates, another line OY perpen- 
dicular to OX (in counter clock wise direction) is 
called Y axis. In cartesian coordinate location of a 
point P is indicated by its distance x from 6 along 
axis (x coordinate) and distance y in direction of 
y axis (y coordinates). Distance in the direction 
OX' and OY' are negative. (Figure 3 a). 

In polar coordinates, location of a point P is 
indicated by its distance r (always positive) from 
origin O and the angle 6 made by OP with OX 
in counter clockwise direction. (r,0) indicate posi- 
tion of any point in space (Figure 3b) 

Conversion from one system to ofher is not 
difficult. 



82 



Siddhanta Darpana 



-r - jc + y 2 x - r cos 

6 = tan -1 y/x y = r sin 

For example, if Bhubaneswar be origin, then 
location of Puri can be indicated in cartesian 
coordinates as 

40 kms south (x coordinate) 
35 kms east (y coordinate) 

In polar coordinates - 53 kms away (r) in 
direction of 40° (0) from south towards east. 

In a plane, two quantities called coordinates 
are needed to locate a point. In space, 3 quantities 
are needed - so it is called 3 dimensional space. 
In theory of relativity, time is considered fourth 
dimension. An event in world is indicated by 3 
space and 1 time coordinates. Hence world is called 
4 - dimensional space time continum. 

For example, a hill top in Puri can be specified 
by its height from mean sea level, in addition to 
two coordinates of plane. 

Three dimensional coordinates : 





Cartesian Space Co-ordinates Spherical polar Coordinates 

F| fl- *a Fig. 4b 

Cartesian space coordinates are measured 
along mutually perpendicular X,Y,Z axis. If a right 
hand screw is rotated from X direction to 'Y 
direction, it will move in Z direction. The distances 
of any point P from origin O along the three axis 
are called (x,y,z) coordinates. 



Calculation at Different Places 83 

In spherical polar coordinates, distance OP of 
P from origin is r coordinate. Angle 6 between 
plane of z axis and OP with X axis is second 
coordinate. In the plane, elevation of OP from XY 
plane (with line OQ) is called </>. 6 takes values 
from to 2 n or 360 \ <p takes values from - 90° 
to + 90° or can take any value. This system is 
more useful for spherical geometry and astronomy. 

Conversion formula - 

r sin 4> = Z, r cos <b cos 6 = x, 

r cos O sin $ = y 

In astronomy, only two angle coordinates are 
used. For places on earth, the distance from centre 
is fixed as radius of earth (r coordinate). OZ is 
line from centre to north pole. Angle 6 is measured 
from prime meridian (great circle or plane passing 
through north pole and Greenwich (London)). In 
India, prime maridian was assumed through Ujjain 
as a reference. O is the angle with equator plane 
(XY plane). In popular terms 6 Coordinate is called 
longitude (- 180° to + 180° and <I> coordinates is 
called latitude (- 90)° (south) to +■ 90 Q (north). 
Positive direction of longitude is called east, and 
negative direction west). 

In astronomy, a second frame of reference is 
also used. This is fixed with refrence to stars which 
don't move. Planet's movement is observed with 
reference to stars. Zodiac or ra& vrtta is path of 
apparent motion of stars in which coordinates is 
measured from 0' to 360°. Deviation from this 
plane is called viksapa or Sara. (-90* to +90*). 



84 



Siddhanta Darpana 



For calculation of eclipse etc, frequently we 
need to convert the figures from equatorial coor- 
dinates to zodiac coordintes. This is called drk 
Karma. 

Sphuta paridhi of earth, at any point is 
circumference of circle on earth's surface parallel 
to equator (latitude) circle or simply called a parallel 
of particular degree. 

(6) Motion of a top and earth's motion 






Spin arround 

axis of the top 

5a 



Spin with steady 
precession of axis 
in a vertical cone 

5b 

Fig .5 



Spin, precession 

and nutation 

5c 



A top rotating fast along its axis stands vertical 
on a rough surface due to gyroscopic stability. Its 
lower end is fixed due to friction with earth and 
it moves away from vertical position and falls due 
to gravity in the end. 

Spin (figure 5a} - Rotation of a top about its 
axis is called spin. When top is rotating very fast, 
its axis is vertical and its appears stationary. 

Precession (fig 5b) - Precession is conical 
motion of the axis of top. Upper point of the axis 
makes a circle about the vertical direction. 

Nutation - When motion of top becomes 
slower, its axis falls further away from vertical and 
rises again alternatively. In steady precession, 
upper point of the top makes a horizontal circle 
on a sphere. In nutation it moves in a wave like 



Calculation at Different Places 85 

th between two horizontal circles on the sphere 

as in fig. 5c. 

(7) Rotation of earth around its axis - Motion 
of earth around its axis is completed in one day 
and causes day and night. Due to that the sphere 
of stars in sky appears to make a daily rotation 
from east to west. This is spin motion of a top. 

Axis of earth is inclined at angle of about 
23-1/2° from perpendicular to the plane of ecliptic 
(i.e. plane of earth's orbit round the sun). Due to 
that the sun appears either north or south of the 
equator. During summer season in north hemi- 
sphere, it will be perpandicular to earth's surface 
at noon time at some place between equator and 
23-1/2° north (Tropic of cancer) 

When the plane containing vertical to ecliptic 
and earth's axis contains sun, inclination of sun 
towards north or south is maximum. These points 
opposite to each other are called summer and 
winter solstice. In summer solstice, axis is directly 
inclined towards sun, and sun is perpendicular to 
tropic of cancer (23 1/2°) 



Autumnal 

equinox 



Summer 
solstice 

N 




Winter 
solstice 



Vernal equinox 



Fig. 6 



At two points on orbit, 90° away from place 
of maximum inclination, the axis of earth is inclined 
side ways and not towards sun. Then sun rays 
*** perpendicular on equator (i.e. on plane con- 



86 Siddhanta Darpam 

taining ecliptic and arth's axis). On such points, 
day and night are equal. 'Nakta' means night in 
Sanskrit, it is called noct in greek. Equinox means 
equal day and night. On one of equinox points, 
sun goes from south to north hemisphece. This is 
called vernal equinox. The other point is called 
autumnal equinox. Northward motion of sun is 
called uttara - ayana and southward motion is 
daksinayana. Both ayanas, make one hayana, a 
complete year. 

Precession of axis - At present, earth's axis 
towards north is directed to pole star (Dhruva 
Tara). So pole star appears to be fixed. Axis is 
moving like precession of a top in conical motion 
due to two reasons - (1) Earth is not spherical, it 
has bulge at equator due to centrifugal force of 
rotation (2) Orbit of moon is inclined to earth's 
orbit at about 5° angle which creates unequal pull 
at different ends of bulge. To some extent, 
inclination of other planetary orbits also affects the 
axis. 

Practical effect of precession of axis is that, 
points of equinoxes move slowly westwards. If 
solar year is counted by motion relative to fixed 
stars, start of seasons shifts slowly. V change of 
equinox, i.e. 1 day change of season occurs in 
about 72 years. One month change is in about two 
thousand years. 

In western astronomy, solar year is counted 
from equinox to equinox. Position of vernal equinox 
is taken as 0° mesa. Difference between vernal 
equinox, and static mesa 0° of Indian astronomy 
is called Ayanamsa. For determining day length, 
rising period of rasis etc, position of sun from 



Qdadati * 1 



87 



eqUhl >ll ? triangle is completed. Since equinox 
SP ^ backward (to west), ayanamsa is added to 
^position. It is caUed sayana sun or any other 

planet. 

Y REFERENCES 

1. For trigonometry, any school text book can 
be referred like S.L. Lone/s Trigonometry. 

2. Cartesian geometry of two dimensiouns can 
be found in any college text book, e.g by 
Loney or by Santi Narayana. Geometry of 3 
dimensions can be found in book by R.J.T. 
Bell. 

3. Results of spherical trigenometry can be found 
in text books by Todhunter or by Gorakh 
Prasad. 

4. Transformation of axis can be found in books 
of classical mechanics or foundations of 
vector/tensor analysis. Differential geometry of 
Weatherburn or by Shanti Narayan can be 
referred for space curves, surfaces and polar 
coordinates. 

5. Polar coordinates/transformation of axis are 
explained in classical mechanics also. 
M.Sc/Hons level text books also discuss mo- 
tion of top. The following books may be 
referred. 

Classical Mechanics - by Goldstein. 



88 



Sictdhanta Darpan 



Principles of Mechanics - by Synge & Griffith 

Mechanics - by Simon 

Earth's top tike motion has been discussed i 
detail in motion of top (4 vols) by W. Sommerfield 
& Felix Klein 

Translation of the text (Chapter 4) 

Verse 1 - I (author) will describe in short the 
various measurements of earth. In second half of 
the book, these will be discussed in detail. 

Verse 2 - Average diameter of earth 
(madhyavyasa) is 1600 yojanas. Multiply this byj 
10,800 and divide by 3,438. You get the paridhi 
(circumference) described in 3rd verse. 

Verse 3-4 - Paridhi at centre (equator) is 
5,026/10 yojana. Jya of 90° is taken as 3438 kala. 
Hence, sphuta bhu-paridhi is obtained by multi- 
plying, madhya paridhi by lamba jya of the place 
and dividing by 3438. Otherwise, this madhya 
paridhi can be multiplied by 12 and divided by 
visuva karna. 




RQ. 7 



Derivation - (1) NS is line joining north and 
south pole. O is centre. The circle perpendicular 
to NS line is called sphuta bhu paridhi. Largest 
circle passes through centre 0, at point A and is 
called equator. Sphuta paridhi at point P is to be 
calculated. 



addition at Different Pious 89 

OA = OP = radius R of earth 
Paridhi at centre is 2 tz R = C 
Latitude of place P is Z.POA = (Aksamsa) 
Lamba amsa = 90° - 6 = ZPOD = <D 
For circle of sphuta paridhi at P, r = DP = 
OP sin 3> 

or r = R sin O 

Circumference = 2tfr = 2jrRsin<l> 

Lamba jya 

3438 



=C sin <P = C 



RsinO 



= C x 



R 

(2) Second 

method is based on 
measurement of 

palabha explained in 
Triprasnadhikara. On 
Visuva samkranti, 
sun rays are perpen- 
dicular on equator, 
i.e. paralled to OA. At 
point P, a pole PR is 
kept vretical of 12 unit lengths. Its shadow PC on 
horizontal surface is palabha and RC is Pala Karna 
or visuva karna. 

In Fig 8, OPR is straight line, RC I I OA or 
RPC and ODP are similar. 




Fig. 8 



Hence 



RC 
OP 



or r = 



PR 
PD 
12R 



or 



Visuva Karna 12 

• * ^ 

R ■ ~ r 



hence (he result. 



Visuva Karna 

• ■ 

Verse 5 - Lanka, Rohitaka, Avanti, Kuruksetra 
etc. are on the prime meridian line (Pradhana 
madhyandina rekha) which passes through both 
merus. * 



90 Siddhdnta Darpawt 

Note (1) Rekha is a straight line in a plane 
but it is arc of a great circle in a sphere (the circle 
passing through centre of sphere, which is 
greatest). Like straight line of a plane, it is the 
shortest distance between two points, and doesn't! 
change the direction. ,| 

(2) . This verse means same as verse 62 in \ 
madhyamadhikara of Suryarsiddhanta and repre- 
sents the convention of treating the longitude 
through Ujjain as reference line (0° longitude). At 
present, the meridian passing through Greenwich 
is 0° meridian. 

(3) According to historical traditions, 'Polaris 
narua' (meaning Paulastya nagara) in present Sri; 
Lanka was the capital of Lanka. However, for 
astronomical purpose, Lanka is the imaginary point 
of intersection of longitude through Ujjain and 
equator (i.e. middle point of that line between 
south and north pole). Lanka is nearest land mass 
near the point; hence it is called Lanka (presump- 
tion) 

(4) Location of original Kuruksetra is not 
known. If present Rohataka (a district headquarter 
in Hariyara) is taken as Rohitaka, then it is 8 pala 
east from madhya rekha. Hence, Bhaskaracarya has 
not indicated it on madhya rekha. He says that 
this line touches regions like Kuruksetra etc. 

Verses 6-9 - Desantara is the east west distance 
between two places with same aksamsa on sphuta 
bhu paridhi (local latitude circle perpendicular on 
polar axis or parallel to equator). 

Multiply this desantara yojana by 60 and 
divide by spasta bhuparidhi. Alternately, multiply 



Calculation at Different Places 91 

by visuva karna in liptikas and divide by 60, 314. 
You will get desantara in danda etc. 

All days, months and years start with mid- 
night at Lanka i.e. from midnight at places on 
madhyandina rekha. If a place is east from rekha, 
add the desantara (ghatl ) to get the midnight time 
at that place, from which day, months will start 
at that place. If the place is west from rekha, 
desantara is to be deducted. 

Derivation - (1) Earth rotates with uniform 
speed around its axis or in the direction of 
bhuparidhi. Complete rotation of bhuparidhi takes 
60 danda or 1 day. Thus by ratio and proportion 

Desantara in danda 60 danda 



Desantara in yojana spasta bhuparidhi 
or Desantara danda = 
Desantara yojana (east west distance) 
sphuta bhuparidhi 

(2) Visuva Karna - I*alabha is length of the 
shadow of a vertical stick (cone or Sanku) at noon 
on a day when day and night are equal. Height 
of sanku is 12 angula. 

Visuva karna or pala-karna is the length of 
hypotenus, i.e. distance from tip of 12 angula sanku 
to the tip of shadow. 

Palabha or pala karna gives a measure of the 

angle of latitude (aksamsa) as sun is vertically above 

equator on visuva day (when day and night are 
equal) 

In Figure 9, X is a place on aksamsa 6\ Angle 
of sun rays at mid day will be 0° at equator, ^o 



92 Siddhdnta Darpana 




12 anguia 



Fig. 9 

it will be 6° at latitude 0* (Derivation 2 after verse 
4, Fig 8) 

i.e. AXYZ = 6 (aksamsa) 

XY is Sanku of 12 anguia (units) of length. 
XZ is palabha and YZ is palakarna. 

Sphuta bhuparidhi = 2 jt r (r = sphuta Trijya) 

= 2 jt R cos 9 (R = radius of earth) 

XY 

= Bhuparidhi x ^r= 

__ Bhuparidhi x 12 anguia 
~ Palakarna anguia 

desant ara yojana x 60 
DeSantara dan^a = sphuta bhOpandh. 

desantara yojana x 60 

= *-f — — x palakarna 

Bhuparidhi x 12 v 

desantara yojana x palakarna in lipta 

60314 

(As per verse 2, bhuparidhi x 12 = 5026/10 
yojana x 12 = 60314 yojanas) 

Verses 10-11 - Some astronomers opine that 
day starts everywhere from the sunrise at Lanka. 
Due to that confusion, the author decides that at 
any place the lord of vara will be ruling from 
sunrise at that place for period of 60 dandas. 

Verse 12 : Bala (power) of yama and yamardha 
is not connected to siddhanta (astronomy) it is 



-A 



Calculation at Different Places 93 

useful for phalita (astrology only). So it is not 
discussed here. 

Verse 13-14 - Bhaskaracarya (and his followers) 
assumes start of all (motion of planets, day etc.) 
from sunrise at Lanka. Thus the ahargana according 
to his theory is different from other theories. This 
separate ahargana (of Bhaskara) doesn't give 
position of planets as they are actually seen, hence 
it is not followed in this book. 

Note - Bhaskara ahargana will give correct 
position of planets for sunrise at Lanka only. Since 
day length is different for different latitudes, 
sunrise will be at different times on same longitude 
also. But midnight will be at same time on the 
whole longitude, hence it gives correct result. 

Verse 15 - Method of finding midnight 
position of planets at ista (desired place) - Multiply 
desantara kala (in danda) of the place with dainika 
gati of graha and divide by 60. Add the result to 
the graha at Lanka at midnight if the place is west 
from Lanka. (Since earth rotates in east direction, 
midnight will be later in a place to the west and 
in the extra time, the graha will move further). 
Deduct, if the place is towards east. 

Verse 16 - Alternately, difference in grahagati 
can be obtained by multiplying dainika gati with 
desantara yojana and dividing by sphuta 
bhuparidhi. 

Note : Desantara ghati of a western place is 
the time taken by earth to reach midnight position 
for that place. Alternate method follows from 
methods of hndig desantara ghati (vrse*9). 



i 






i 



94 Siddhanta Darpam 

Veise 17-21 - Old method of finding longitude 
- calculate the time of purna (full) candra grahana 
(lunar eclipse) at madhya rekha (prime meridian 
through Lanka or Ujjain). 

(Note - Exact time of Purna grahana is the 
time of unmilana (when moon starts emerging from 
shadow). 

By observation, see the actual time of Purna 
grahana at your place. The difference in time is 
desantara kala. 

If the place is west from Ujjain, then the time 
found by observation (drk-siddha or vedha) wul 
be less than calculated time (i.e. eclipse will be at 
same time, but corresponding local time will come 
later at western place). For places east of Ujjain, 
observed time will be more. 

Time difference can also be calculateed on 
basis of sparsa (when moon starts entering the 
shadow) or moksa (when moon completely emerges 
from shadow). 

To find desantara yojana, multiply it 
(desantara kala) by sphuta paridhi and divide byj 
60 (already explained in verse 9). j 

To calculate graha at ista time, multiply the] 
dainika gati of graha by ista kala and divide by 
60. Add the result to graha at midnight at the 
place. 

Notes : (1) Time difference (in danda) frorr^ 
Lanka midnight is due to two components - (l)i 
difference between midnight times at the place ancl 
at Lanka (2) Time lapsed after midnight of the 
place at desired time. 



95 



Calculation at Different Places 

Dainika gati of graha is movement in 60 danda 
(1 day). Hence movement in ista danda is 

Dainika gati x ista danda (kala) 



60 



components of ista kala are added or sub- 
stracted as explained before. 

(2) Candra grahana is due to covering of moon 
by shadow of earth, both of which are at one 
place. Thus there is no parallax and it is seen 
similar from all positions. But Surya grahana is by 
obstruction of sun's vision by moon (at i/400 of 
the distance). Their relative directions are seen 
different from different places, (called parallax), 
hence surya grahana starts at different places at 
different times. Hence only candra grahana can be 
used for comparison of midnight times. 

(3) Terms of grahana 




Fig. 10 

In fig 10, S is sun, E is earth and M is orbit 
of moon. C is shadow cone of earth due to rays 
from sun. 1,2,3,4 are successive positions of moon. 

1. position of moon touching the shadow - at 
sparsa kala. 

2. position of moon when it has just entered 
completely in shadow - Nimilana or sammilana 
(meaning closing of eyes) kala • 



% Siddhanta Dnrpana 

3. Position of moon about to emerge from 
shadow; unrmlana (opening of eyes) kala 

4. position of moon when it has just emerged 
completely from shadow - Moksa kala 
Grahana will be discussed more completely in 

chapters on candra and surya grahana 

(4) Other methods of finding longitude - Now 
very accurate watches are available and any event 
can be observed with telescope more accurately. In 
observing candra grahana, there will be difference 
of 2-3 minutes in observation by different persons. 
Eclipse of satellites of jupiter occurs daily. It is 
observed through telescope and compared with 
time given in nautical almanc. This will give 
accurate longitude. 

Alternatively, two watches are to be tallied 
with local times of places, whose longitudes are to 
be compared. They can be tallied with sunrise or 
preferably at midnight time. Then by telephone, 
the local time of the two places can be compared. 
The time difference will be desantara kala. Nowl 
T.V. and radio announce Indian standard times 
(mean time at 82° 30' east of greenwich). Local mean 
time can be found by correcting local true time! 
with time equation (fixed for particular days ofj 
solar year or sun position). From that time] 
difference, difference with 82° 30' longitude can be 
known. 

(5) Time can also be known accurately b 
movement of stars during night. This is particular]: 
useful for sea journeys in a clear night. Since, 
method .of finding longitude was known since 
remote past in India, long journey in sea wa 



Calculation at Different Places 97 

possible. Due to difficulty in knowing time in 
absence of watches, this method could be known 
in western astronomy only in 1480 A.D. after which 
Cobumbus could undertake his journey, in 1492 in 
pursuit of sea route to India from Spain. Finding 
latitude is easy through palabha, discussed in more 
detail in Triprasnadhikara. 

Verses 22-24 : By above corrections for 
desantara kala, we get the graha for niraksodaya 
kala (sunrise time at equator at same longitude). 
Due to difference in aksamsa (north south distance) 
from Lanka, cara samskara is needed, because 
sunrise times are different for different places on 
same longitude due to aksamsa. 

From sphuta ravi (sun) kranti (true inclination 
of sun from vertical in north south direction i.e. 
inclination from vertical at noon), find cara danda 
(time in danda by which day-half is longer than 
normal day half of 15 danda). Multiply it with 
dainika gati of graha and divide by 60. If sun 
(sayana) is in six rasi from tula to mina, add the 
result to the position of graha. If sayana sun is in 
mesa to kanya, then deduct the result. For finding 
graha at the time of sun set, do the reverse process. 

Notes (1) This part (chapter 1 to 4) is 
madhyamadhikara, dealing with mean position of 
planets. Nothing has been so far discussed, as to 
how, true (sphuta) position of planets can be 
found. Sphuta kranti of sun can be found only at 
moon time by direct observation. By comparison 
with previous days kranti, it can be calculated for 
sunrise time (3/4 of the difference of 1 day kranti 
will be added to previous noon figure? to find 
kranti at sunrise). 



98 



Siddhdnta Darparta 



2. Mesa to kanya - 1st six rasis are in north 
hemisphere and other six are in south (sayana rasis 
to be more accurate). When sun is in southern 
hemisphere, days will be smaller in north hemi- 
sphere compared to night. Hence sunrise will be 
later and sunset earlier than equator (where day 
night are always equal) Thus graha will move for 
more time at sunrise compared to sunrise at 
equator, difference of motion will be added. 

3. Cara is variation of day from 30 ghatika, 
caradala is half of cara. In short cara is used for 
caradala which is directly calculated. Jya of cara 
(angular difference in earth's rotation) is called cara 

jya. 



V A 




Fig. 11 

4. Explanation of cara - (difference in day, 
length) O is the place for which it is to be found 
out for how long, a graha will be above horizon.; 
NOS is North south line (ksitija rekha) POP' is the 
north south line at equator (Z.PON is equal to 
aksamsa of O). 

NPVSP' is yamyottara vrtta, i.e. the vertical 
circle in the plane of longitudinal circle (great circle 
passing through north pole and vertical at place 
O). 



Calculation at Different Places 99 

A planet in kranti vrtta appears to move daily 
• a vertical circle at equator in east west direction. 
Its diameter BOB' is perpendicular to north south 
r e P'O P at equator. This circle is called ahoratra 
vrtta (only diameter is seen in perpendicular^plane). 
Corresponding to point O, the planet rises in the 
east goes upto B, highest point in sky (south from 
vertical in north hemisphere) and sets in west again 
at O. Motion from O to B' and back to O are not 
visible as these are below the horizon. Both motions 
OBO or OB'O take 12 hours each. 

CK'C is the diameter of ahoratra vrtta (diurnal 
circle) of a planet in south hemisphere. At equator, 
it is visible for motion K'C K' for half the day i.e. 
12 hours. However, at place O, it rises only at 
point M' and is not visible for period K' to NT (in 
12 hours) which is called car a. 

Time for K'C = 30 ghati (12 hours) 

For K'M' in morning and M'K' in evening, 
sun (or a planet) will not be visible above horizon. 

Thus length of the day is 30-2 K'M' 

A MA' is diameter of ahoratra vrtta of a planet 
in north hemisphere. 

Kranti of planet corresponding to AA' is AB 
(north) and corresponding to CC it is BC (south) 

Carakala is time corresponding to movement 
between KM or K'M' (called ksitijya or kujya) 

Radius of ahoratra vrtta is called dyujya ('Dyu' 
means light) 

Carajya of planet is projection of kujya on 
visuva vrtta BOB'. It is OR' for north kranti and 
OR for south kranti. 



100 



Siddkdnta Darpanw 



Angle made by carajya (length of circum-! 
ference) at the centre expressed in prana is called! 
caraprana or carakhanda. 

5. Methods of calculating carajya 

(Chapter 6-104, p-352) 




12 Artgula 



Fig. 12 



AB is cone at a place with aksamsa 0°. It ii 
kept vertical on day of equinox at noon time. Sino 
sun rays are perpendicular to equator or that day; 
it will make angle 0° with AB. 

BC is shadow at that time (figs 12) 

L BAG =0 

Length of AB is 12 angula as per convention 
BC is palabha. 



tan0 = 



BC Palabha 



■(1) 



AB 12 

Now according to figure 11 in para(4), BA i 
north kranti. ZBOA is angle of kranti (angle no 
shown) 

AL is kranti jya (AL J_ OB) 
AL = OK 

Now ^KOM = = aksamsa 
KM Ksitijya 



Tan 6 



OK 



From (1) Tan = 



Krantijya 
Palabha 



(2) 






:\ 



12 



Calculation at Different Places 101 

Krantijya x Palabha 
so, Ksitijya = ^ (3) 

PKO and PMR' (grand circles) are both 
perpendicular on AK and BO. Due to similarity of 
s D herical triangles (as in plane triangles) 

AE = BO 

KM OR' 

BO x KM _ Ksitijya x Trijya 
or carajya OR' = —^ - ^7 

(6) The difference in planet motion at sunrise 
is calculated by proportion of motion in carakhanda 
compared to dainika gati in 60 danda. 

Verse 25 - The value of cara danda for a 
particular sphuta surya previous year will be same 
for the equal rasi of madhyama surya this year 
(exactly same for equal sphuta surya). This ap- 
proximate equality is used for checking the results 
obtained through palabha. By taking this value of 
cara danda, there will be negligible error. 

Verses 26-30 - Bhujaphala samskara - Now, I 
tell about another samskara (correction) in madhya 
graha based on niraksa lagnamana and ayanamsa 
etc. Mid-night calculated from madhya ravi is 
different from midnight of sphuta ravi. Difference 
between sphuta and madhyama ravi is called 
bhujaphala and correction for that is needed. 

Add ayanamsa to madhyama ravi, find manda 
bhujaphala, multiply it by udayasu (time of rising 
of rasi in prana) of the rasi at equator (niraksa) 
and divide by 1800. Multiply the result by dainika 
gati of graha and divide by asu of madhya ravi 
savana dina. The result in kala etc is to be added 
°r substracted from madhya graha for bhujaphala 
samskara. (There are 21659 asus in a madhya 
savana dina). 



102 Siddhdnta Darpana 

For correction in slghraphala, mandocca of 
candra or bhujantara of rahu, reverse is done, 
(positive bhujaphala is to be substracted or vice 

versa) 

Notes (1) Manda bhujaphala is neither ex- 
plained nor method of finding it has been described 
in madhyamadhikara (chapters 1 to 4). 

Manda bhujaphala is the correction to graha 
rail due to its unequal speeds which is slowest at 
mandocca. (Since sphuta graha is closer to man- 
docca than madhya graha, it is termed as attraction 
of mandocca). 

Real motion of earth E is in an ellipse around 
sun S at one of the focus. The farthest point E on 
far side of major axis is the slowest point called 
mandocca. (It is manda = slow and highest = Ucca ) 
E, is closest to sun called the nica point. Midd e 
points of the orbit Ex and E 3 are not at right angle 
to direction of major axis but towards mandocca 
position (apparent attraction towards it). 

Apparent elliptical motion of sun around earth 




Fig. 13b 



is explained by combination of two circular move- 
ments. Fig. 13(a) is real orbit of earth round sun 



Calculation at Different Places 103 

Fie 13b indicates apparent positions of sun calcu- 
lated by combination of two circular motions. E is 
earth around which madhyama surya M is moving 
in a circle in anticlockwise direction. 8 positions 
are indicated as Mi, M 2 — M 8 . Sphuta graha S is 
rotating in a smaller circle (manda paridhi) in 
opposite direction. Both complete the rotation in 
equal time. Corresponding positions of sphuta 

graha are indicated by Si, S2 S%, 

At position 2 for example ZS 2 M 2 V 2 = Z-M{E M 2 
as speeds of madhya graha and manda graha are 
equal. Apparent position K 2 on kaksa vrtta is 
sphuta graha. M 2 K 2 is called manda phala. S 2 V 2 
perpendicular on manda trjya is called manda bhuja 
phala (fig 13 c) which is almost equal to man- 
daphala as mandaparidhi is very small compared 
to madhyaparidhi. M 2 V 2 is kotiphala. 




Fig. 13c 



Mandaphala and bhujaphala is negative in 1st 
semicircle after mandocca. (it is to be substracted 
from madhya graha). In 2nd semicircle it is positive. 

Kaksa vrtta is 360° or 21,600 kala. 

Manda paridhi is expressed in angle in 
proportion to length of kaksa vrtta. 
Sin L$ 2 M 2 V 2 = Sin LM& M 2 

or - V * s Ul ft 

S *M 2 *" EM 2 



104 Siddhanta Darpana 

S2 M2 (Manda Trijya) 
Bhujaphala S2 V 2 = M2 P2 x ^ (kak§a Trijya) 

man daparidhi (sphuta) 
= Bhuja jyS x ^ " 

m andaparidhi 
Kotiphala = Kotijya x ^j 

Mandaparidhi also changes slightly, because, 
earth is not at centre of orbit, but on one side at 
the focus. 

(2) Udaya kala of different rasis is calculated 
in chapter 6-121. Due to oblique direction of rasis 
with equator (24- or 23 '27' more accurately), the 
time taken by different rasis to rise is different. 
As we move away from equator this inclination 
with local horizontal plane increases. Difference in 
rising time of rasis becomes more. However, total 
time of rising of all rasis will be same as Naksatra 
dina for all places. The rising time Of rasis for 1st 
to 6th rasis is same as that of 7th to 12th rasis m 
reverse order. At equator, position of 1st to 3rd 
rasi is same as 6th to 4th rasi (symmetric for sayana 
rasi), hence their rising times are same. For 
difference in start of mid night at ^uator, only 
the rising times at equator are needed. A com- 
parison of traditional rising times based on Surya 
siddhanta and modern values is given below- 



.'< 



i 



Sayana rasi 



Mesa 



Vrsa 



Kanya 



Simha 



Tula 



Surya siddhanta 
Parama Kranti 24* 



Mina 



Asu 



Vrscika 



Mithunal Karka 1 Dhanu 



Kumbha 



Makara 



1670 



1795 



Fala 



278 



299 



Minutes 



New observations 
Par ama Kranti 23*27' 

Asu |Pala|Minutes 



111 



120 



1935 I 323 I 129 



1675 



1794 



1931 



279 



299 



322 



111.7 



119.6 



128.7 1 



At other places, udayasu of rasi is lessened 
by carasu. It is added for rasi 4 to 9. 



Calculation at Different Places 105 

(3) Since Udayasu is calculated for naksatra 
dina and dainika gati is calculated as per savana 
dina of 21659 asu. 

au dainika gah 

gati in 1 asu = 21659 

ud ayasu x dainika gati 
gati in udayasu = 21 659 

Hence correction for manda bhujaphala 

manda bhujaphala X gati in udayasu 

= '~ 1800 ~~ 

*. because udayasu is for rise of 30° i.e. 1800 

kala. 

Verse 31-32 - Alternate method for bhujantara 

sanskara / . 

Ravi manda bhujaphala x dainika gati 
Bhujantara = ^ 21,600 : 

= mandabhuja phala -^ (21,600 -5- dainika gati) 

It will be added or substracted as before. 

Note - In this formula, different rates of rising 
of rasis and difference between naksatra dina and 
savana dina are ignored. 

Verse 33 : After, bhujantara samskara, I am 
telling the method of udayantara samskara which 
is due to difference between madhyama ravi in 
kranti vrtta and imaginary madhya ravi in nadivrtta 
(in plane of equator). 

Verses 34-37 - For this purpose (for 
udayantara samskara) make the madhyama ravi 
sayana (add ayanamsa). Find the bhukta asu of 
that rasi (part of udayasu of rasi in proportion to 
lapsed degrees in that rasi). Add the udayasu of 
previous rasi starting from mesa. Then calculate 
the kala of sayana ravi and substract from 1st result. 



206 Siddhanta Darpana 

Multiply the difference by the dainika gati of graha 
(in lipta) and divide by 21,659 (as dainika gati is 
for savana dina of 21,659 asu) The result is 
udayantara phala. Substract the result from madhya 
graha, if ravi is- in sama or even pada (2nd or 4th 
quadrant) and add if ravi is in vteama pada (1st 
or 3rd quadrant - to 90° or 180' to 270° from 
mandocca). For correction in pata or ucca, do the 
reverse. 

Notes : (1) Madhyama ravi + Ayanarhsa = 
Sayana madhyama ravi = S 

Bhukta asu for S = rising times for rasis from 
0° to S 

1 asu time = time for movement of 1 kala at 

equator 

Hence Bhukta asu of S = Its kala at equator 

= E 

Kala of S = S' 

Correction for observation in plane of equator 
= E-S' in kala equivelent to asu time. 
Difference in madhyama graha = 

Dainika gati 
(E-S) x 21659 

as 1 day is of 21659 asu (savana dina) 
This difference is negative for 1st and 3rd 
quadrant i.e. Sayana ravi is more in kranti vrtta 
than in nadivrtta. At 90° and 270° they are equal, 
and no correction is needed. 

(2) This is effect of transformation of coor- 
dinate axis from ecliptic to equator, because time 
is measured by movement along equator (asu is V 
movement) . 



Calculation at Different Places 107 

Verse 38-39 - The three samskara (cara, 
hhuiantara and udayantara) can be made to sphuta 
Taha also instead of applying it to madhya graha. 
Then we will use sphuta graha instead of madhya 
ha in a ii places. Once the samskara has been 
done to sphuta graha, it is not to be applied again 
to madhya, mandocca and sighrocca because these 
results are used to calculate sphuta graha. 

Note : The sanskaras are for difference in time 
measurements and not due to madhya or sphuta 
graha, hence correction to any value can be done. 
In short, correction time difference between mad- 
hya and sphuta is negligible. 

Verses 40-41 - If we take asu arising out of 
mandaphala of ravi while making bhujantara 
samskara, then udayantara karma is done from 
sayana madhyama ravi. 

When we take asu equal to kala of man- 
daphala of ravi then udayantara will be done from 
sayana ravi before bhujantara sanskara, both are 
to be done separately. 

Note : (1) In taking asu equal to mandaphala, 
it is already converted to value in equator plane; 
hence separate udayantara samskara is not neces- 
sary. 

(2) A review of all corrections - (a) Desantara 
sanskara - It is due to different times of sunrise 
which is earlier in east. Hence time in east is more 
counted from sunrise or midnight. At present 
reference is not 0° longitude only. Every country 
has fixed reference time according to time zone 
from 0° longitude through Greenwich. Thus Indian 
standard time is standard time for 82 * 30' east of 



108 Siddhanta Darpana 

greenwich, i.e. 5-1/2 hours more. Correction for 
local standard time is done for difference in 
desantara (longitude) Since 360° rotation of earth 

24 X 60 . . . . 
in 24 hours/ 1° rotation is in 36Q = 4 minutes. 

Hence 4 minutes time is added for each degree 
longitude towards east. 

Local standard time - Indian standard time 
= (longitude - 82° 30') in degrees x 4 minutes 
(b) Cara sanskara - Midnight or midnoon is 
same for all places in a longitude. When time is 
measured from midnight (in hour system), then 
no correction is needed. However, in India, savana 
dina is counted from sunrise which is different at 
different latitudes. Difference in day length in- 
creases as we move away from equator. In practice 
we do not correct the time, but find the time of 
sun rise. Time of sunrise depends on position of 
true (sphuta) sayana sun which is fixed for a 
particular day of a solar year like christian era. It 
also depends on latitude of the place. Thus date 
wise charts are prepared for sunrise time at 
different longitudes (in local mean time), at 1° or 
10° intervals. It can be calculated from kranti of 
that day noted from pancanga or calculated from 
sayana ravi. • 

Difference between true time and sunrise time 
- both counted from midnight gives ista kala in 
Indian system. 

(c) Bhujantara sanskara - This is due to 
difference in standard time and true time - both. 
Local standard time is calculated on the assumption 
that each day is of 24 hours. Day length is made 



Calculation at Different Places 109 

< two components. To move from 1 naksatra to 
irnaksatra again it takes 23 hours 56 minutes 
a to earth's daily motion. Meanwhile, sun also 
^rfves about 1° ahead due to orbital movement of 
Sin S ame direction (360° in about 360 days). 
To cover that distance more earth takes about 4 
minutes more (360° is covered in about 24 hours). 
Thus naksatra dina is 23 hours 56 minutes = 21,600 
asu and savana dina is 24 hours = 21,659 asu. 
Difference is 59 asu = about 4 minutes (= 60 asu). 
While naksatra dina is fixed, extra 4 minute 
component varies and each savana dina is not 24 
|purs exact. But the watches are calculating 24 
hours for each day according to standard time. The 
Standard or mean time and true or solar time start 
together at sayana mesa sankranti, 23 March., when 
*Jay and night are equal. Around 24th April when 
sun is at farthest (mandocca is at nirayana mesa 
§0° or sayana 32°), sun is slowest. So days are 
|*naller than 24 hours after 23rd March. By taking 
|4 hours for each day clock time is slower than 
true time. This addition in clock time to get true 
time accummulates for about 6 months upto 14 
minutes. Then it is negative correction and again 
both times tally on 23rd March next year. 

Effect of 4 minutes shorter naksatra dina is 
that a particular lagna (e.g. mesa) will start 4 
minutes earlier on next day. Effect of difference in 
true time and standard time is that sun will be at 
top most position at true noon not at lqcal mean 
noon (12 hrs local standard time). This is also called 
correction due to time difference, or velantara 
sanskara. The formula for knowing difference in 
true and standard time is called time equation. 



i^'i : 2 



m\ 



110 Siddhanta Darpana 

This difference depends only on sun's position 
(indicating bhujantara) or the day of solar year. 

(d) Udayantara sanskara - This is negligible 
and is not necessary when bhujantara is measured 
in asus. In modern astronomy also, this is included 
in time equation. 

Verse 42 - Multiply aksamsa kala by 
bhuparidhi and divide by 21,600. Then we get the 
distance of place from niraksa (equator) towards 
north or south on the yamyottara vrtta (longitude 
line). 

Note - Bhuparidhi covers 21,600 kala. Aksamsa 
kala is north south distance from equator in kalas. 
Thus the distance from equator is calculated 
because 1 kala is same on longitude line or equator. 

Verse 43-45 - To save enormous labour in 
calculating graha, I am giving 'padaka' of surya 
etc. like Kocanacarya. ('Kocanna' was an 
astronomer of Andhra Pradesh who had prepared 
charts for easy calculation. These charts were 
popular in south Orissa also at the time of author). ^ 

Ahargana is given for years 1,2 — — , 10,20, — , 

100, 2000 , thousands, lakhs, ten lakhs, crores 

and ten crores. These start from madhyama surya 
at mesa sankramana. Vara Suddhi has been done 
in this. To calculate the ahargana, add the figures 
.given in table and divide by 7. If correct vara 
doesn't come, then add or substract 1 for tally with 
vara. 

Verses 46-51 - By this method, ahargana for 
first day of panjika is calculated. Graha is calculated 
for that ahargana from their respctive padaka 
(tables). In this addition, we take figures upto 5 



-J4- - 

5r-. : . 




OOculation at Different Places 111 

divisions from rasi (para). By this, graha gati can 
£e calculated for up to 1 arbuda (10 8 ) days. 

After writing padaka of graha and ucca etc, 
their dhruva (starting position) at beginning of kali 
and beginning of Karanabda (standard year for 
start of calculation by author-mesa samkranti of 
1869) are written. Also write the dainika gati of 
graha, ucca and pata. Write bhujantara, cheda 
(part) mandocca hara (part), patahara and desantara 
fcala etc. In the 73 tables, while adding rasi etc of 
aha, multiples of 12 rasi (1 revolution) are 
jducted. When calculation is from kali beginning, 
~§ge get madhyama graha etc for Lanka midnight. 
|f calculation is from karanabda; then value is for 
sunrise. For madhyamana of candrapata (rahu), the 
angles are deducted from dhruva. Result is 
deducted from complete revolutions. 

Verses 52-55 - After obtaining madhyamana, 
fle§antara samskara etc. are done. Then 
ahasphuta is done with help of table of 
landaphala. As a siddhanta grantha, the tables 
should have been given after their related text. But 
at the time of printing (in 1899) all charts were 
#ven in appendix. 

Verse 56 - For convenience in grahasphuta, I 
(author) have given phala, dhruva, gati etc in chart 
for 1 to 10 8 days. After calculating graha sphuta 
according to charts, you may not observe the graha 
m same position. Then correction is to be made 
y seeing dainika gati, dhruva padak etc. in second 
P«t of this book. 

in u ISe 57 " k* a y tne ^ orc * Jagannajha reside 
m y heart who is worshipped by Kubera's friend 




* 22 Siddhanta Darpana 

Siva at Nflacala situated at aksamsa 4/27 (palabha) 
and desantara 8434 viliptas which are 4/19 palabha 
and 9138 vilipta according to new calculations. 

Verse 58 - Thus, ends this fourth chapter 
written by Sri Candrasekhara born in renowned 
royal famfly of Orissa. Siddhanta Darpara is for 
tally of calculation and observation and education 
of students. Padaka charts have been given for fast 
calculation. 



1 



i 



t$ 




■ 'i 



B. SPHUTADHIKARA 

Scope - This part deals with finding true 
position of planets. So far we have calculated 
methods of finding mean position, which assumes 
constant average speeds of planets, to a first 
approximation. This part contains two chapters. 
Chapter 5 discusses true friotion of planets. Chapter 
6 deals with special corrections to moon's motion 
and accurate pancanga on that basis. 



Chapter - 5 

TRUE PLANETS 

(Making grahas sphuta) 
General Introduction 
(1) Concepts of Plenetary motion 




Figure 1 (a) 




Figure 1 (b) 



114 



Siddhanta Darpana 



Copernicus 

S represents the sun 
and P the planet The line 
KL turns with the angular 
velocity of the planet round 
the sun, while LP turns at 
twice the rate. The length 
KL and KS is specified for 
each planet. (Not drawn to 
scale) 



Ptolemy 

£ is earth, P is planet 
Line A L turns with mean 
angular velocity of the 
planet round the sun. The 
line LP Turns with the 
mean angular velocity of 
earth. Length AC = CE is 
specified for each planet 
Length LP is related to the 
earth sun distence (for outer 
planets) 

From the data collected over centuries, ap- 
parent circular motion of planet with some loops 
and retroacting motion were detected, where it was 
difficult to find a pattern. But Hipparchus (100-120 
BC) and Ptolemy (85-165 AD) were able to describe 
it on the basis of epicyclic moitons. As explained 
in diagram of Ptolemy (Fig la) planets moved in 
circles whose centre moved on some other circle 
round the earth, centre of this circle was slightly 
different depending on changes in the velocity of 
planet. 

This was successful in predicting the future 
position of planets, but was unable to reveal any 
law of nature. Copernicus modified the pattern 
with similar construction; (Fig lb) but with sun at 
rest in which patterns were easier to detect. Based 
on this construction, Kepler (1571-1642) framed 3 
laws- 

P 




Finurft 2 - Kpnlarian Orbit 



True Planets H5 

Line SP joining sun S to a planet P sweeps 
out equal areas in equal time intervals (rule 2). P 
moves on an ellipse with S at focus, (rule 1). OA 
is semi major axis and ratio OS/OA is called 
eccentricity. 

Third law is that square of time period T of 
revolution of a planet is proportional to cube of 
its mean distance from sun. 

These laws led Newton to prove that all 
matters attract each other with a force proportional 
to inverse square of the distance between them. 
Together with plausible assumption that force is 
proportional to masses of attracting matter, it 
formed his theory of gravitation. 

However, the method of calculation of 
planetary position remains the same. In both the 
methods, we calculate the direction and distance 
of planet from sun (heliocentric position). Then on 
basis of earth's distance and direction from sun, 
we calculate the direction of planet from earth 
(geocentric position.) Heliocentric position is only 
a mathematical necessity. Actual observation is 
always from earth, equal to geocentric position. 

(2) Calculation of planets in Western 
astronomy - Calculation of sun's position is 
simplest. We calculate position of apsis (nearest 
point on major axis — Indian method starts with 
farthest point) mean anomaly (angle with apsis) 
a nd position of vernal eqinox from which lon- 
gitudes are measured. In a solar calender, sun's 
revolution is almost equal to year and position, 
longitude and latitude of sun depend on date of 
calender with minor corrections. * 



116 I 

Siddhdnta Darpand 

* 

Moon's orbit has perturbations due to attract 
tion of sun and other planets. Movement of its 
node is faster (Due to its nearness to earth and 
effect of sun, parallax etc, its accurate calculations 
for eclipse is needed. First, we derive the formula 
for calculation. 

To know the true position - (1) A planet mi 
its etiptical orbit with sun at focus is calculated to 
know its direction and distance from sun 
(Heliocentric position) 

(2) Position of earth is calculated from sun.? 
From its direction and distance we calculate! 




Figure 3 

direction and distance of planet from earth 
(Geocentric position). 

Explaination of Anomalis (Fig 3) - APA' is 
eliptical orbit of a planet ^nd S is the attracting 
sun at a focus. Revolution of the planet is counted 
from position A when it is closest to sun (near 
end of major axis) After 'd' days planet is seen at 
position P, L ASP is manda kendra = (True 
anomali). 

Auxiliary circle is drawn on diameter AA'. PBi 
is perpendicular on major axis, on extension itj 
meets anxiliary circle on P\ ZAOP' = <p is called 
eccentric anomaly. 6 and <p are measured in length^ 
of arcs (radian measure). If daily mean motion of 
planet is n radians, then 2ji/u is the period of 



„, *c 117 

True Janets 

Virion If daily motion is always n, then the 
SS after 'd' days with be 'nd' which is called 
P pan anomaly (madhyama manda kendra). This 
Sffl be true anomaly, if speed of planet (angular) 
Tmnstant. According to second law of Keplar -- 

lS _ AreaASP = d = d , 2* = dn 

Area of ellipse Time of revolution n 2jt 

Area ASP _ area of ellipse b Jtab 
Area ASF " area of circle a ^ a 2 

where a and b are semi major and semi minor 
axis. 

Area ASP _ Area ASP = Area ASP 
SO Area of ellipse ~ Area of circle jra 2 

But Area ASP' = Area AOP' - Area, SOP' 

a 2 <I> BP x OS _ a 2 ^ _ asinOae 

= T "■ 2 " 2 2 

a * 
= — (0 — esinO ) 

Where e = eccentricity (cyuti) of ellipse. 

dn _ Area ASP _ a^ (<f> - esin<I>) 
Hence — = Me& of eUipse 2 ^2 

or dn = <X>-e sin <I> (1) 

This is relation between mean anomaly and 
eccentric anomaly. 

Relation between true anomaly and eccentric 
anomaly-Polar equation of ellipse is 

a (1 - e 2 ) 
1 + ecosB 

As per difinition of ellipse 
SP =e x distance of P from directrix 

=e x distance fo B from directrix 
(1 to major axis) 



118 Siddhdnta Darpana 

=e (distance from centre to directrix 
- centre to B) 

= e (J - OB) 

a 

= e(- - a cos <X>) = a - e a cos O 
e 

or, radius vector (karna) = a (1 - e cos <p) — (2) 

a (1 - e 2 ) 

So = a (1 - ecos^>) 

1 + e cos 6 / 



Or, 1 + e cos = 



1-e 2 



1. - e cos <b 



a 1 ~ e -, (e cos <p - e) 

or, e cos 6 = — - 1 = - — ~ L 

1 — e cos <p 1-e cos <I> 

cos <I> — e 



or, cos 6 — 



1-e cos <£ 



_ 2 & 1 ~ cos # 1 - ecos <I> - cos <I> + e 

Tan — = — = — 

2 1 + cost? 1-e cos <I> + cos 3> — e 

_ 1 + e 1 - cos<l> 

1 - e ' 1 + cos 

1 + e 2 O 

= _ f an 

1-e 2 

-v/ i+7 $ 

or tan- = V . tan -~- (3) 

2 1 — e ^ 

Equations (1), (2) and (3) can be used to find 
manda kendra (True anomaly), manda karna 
(distance from sun) and d (time in days) from A 
(perihelion-nearest point). 

For practical purpose, these equations are not 
convenient. For calculation on basis of average 
velocities which are known accurately, equation (3) 
needs to be expanded in a power series of small 
e as coefficient of sines of average position. 



119 
Tru e PI*** 

Equation (3) can be re-written on basis of 
formula of Trigonometry 

tane = -^ x e + e -i j ' 

where e = base of natural logarithm 

1 11 

= 1 + -[1 + |2 + 13 + - 
To differentiate it from eccentritity, we write 
is E then (3) becomes _.^ 



i(E2 + E 2 ' i( E 2 



-i* 



E* - 1 _ t/ltc E" - 1 
01 E 1 * + 1 ~ 1 ~ e * E i4> + 1 

Adding 1 to each side and substracting from 



1, then dividing 

vr^r (E i4> + i) - vrr^ (e 1 * " *) 



.„ vr+ r a?* - i) + vr^ ce^ + i) 



E i<D _ vT~T~e - vT~^"e 

= : dk where p = TrvTTlr^ 

1 - pE 

or E = E ^^ 



>i* 



1 - pF 

Taking logarithm of both sides, 

i<9 = i<D + log (1 - pE _i *) - ^g (1 -pE 1 "') 

+ 2! (E 3i^» _ E -3i* } + .... 

3 v 



120 Siddhanta Darpana 

or 

6 = d> + O + tCE 1 * - E- i4> ) + E_ (E 2i4> E" 2i *) 

+ £? (E 3i* _ E - 3i V 

3i 
or 

2p 2 2p 3 

= $ + 2p sin <D + -£- sin2 <D + -*j- sin 3 3> + . . . 

P 2 P 3 

or ^ = +2 (p sin 0> + y sin2 0> + *- sin3 . . . ) 

(4) 

Equation (4) needs to be expressed as a series 
in mean velocity n and d which are easily 
determined. 

For this, we use Taylor's infinite series based 
on Lagranges mean value theorem of differential 
calculus. This is written as 

h 2 

f (x + h) = f (x) + h f (x) + — f" (3^ + .... oo 

let y = x + h f (y) (5) 

Then F(y) = F [x+h f(y)]. Then by Taylor's 



Theorem, 



h 2 d 



= F(x) + h.f(x).F (x) + -j^ Wy)-F (x)] 



[3'dx 2 



[f(y) 3 .F (x)] + ... 



+ S £^ ^ ' (X)1 + ^ 

From equation (1), dn - <S> — e sin O 
or <I> = dn + e sin <I> = m + e sin <J> where m = dn 
This is in form of (5) where y « O, x=m, h = e 
Let F (O,) * <t>„ then F(m) = m and F (m) = 1 



True Planets 

Hence form (6) 



. e 2 <* 2 

<j) = M + esinm.H — 33 (sin m 



[2 dm 



1) 



+ 



L 3 ' dm 



<r 3 

(sin m.l) 



121 



+ 



d 3 4 

(sin .m .1) + 



e 5 d 4 



— (sin m .1) + . . . 



L 4 ' dn 3 |5 dm' 

Expansion of Sin n m is given by 

(when n is even) • 

1 

— — : — [ cos nm - n cos (n — 2) m 

2 n_1 (-if 2 

n (n - 1) n (n - 1) (n - Z) 

+ .- — - cos (n - 4) m - — - — - ' 



sin 11 m = 



12 

cos (n — 6) m + . . ■ . ] 
When n is odd, then 

1 



|3 



sin n m = 



n -1 
2 n_1 (n - 1) — 



r *• . n (n — 1) 

[sin nm - n sin (n - 2) m + — ^— — - sin (n - 4) m 

n (n - 1) (n - 2 ) 

— — sin (n - 6 ) m + ......]. 



Hence - — (sin m ) = 



d (1 — cos2 m) 



dm 



dm 2 

d 2 (3 sin m - sin3m^ 



= sin 2 m 



2 ( sin m ) = 2 
dm z dm 2 

= d (3 cos m — 3 cos 3 m 
dm 




— -7 (3 sin 3m - sinm) 
4 



222 



Siddhdnta Darpana 



d 3 , . 4 , d 3 

^ (Smm) = ^ 

1 2x3^ 
(cos 4m - 4cos 2m + — ^— x 1) 

2 3 (-1) 2 



L2 



- (4 3 sin 4m-4 x 2 3 sin 2m) = 4 (2 sin 4m - sin 2m) 
d 4 



dm 



(sin m) 



dm' 



1 5x4. ^ 

(sin 5m - 5 sin3m + — — sin m) 



2 4 (-1) 



» 

= — (5 4 sin 5m - 5 X 3 4 sin 3m + 10 sin m) 
16 v 



dm" 



6 * 

(sin 6 m) = 



dm 



5 



2 5 (-!)" 



(cos 6m - 6 cos 4m 



6x5 „ 6 x 5 x 4 

+ .. cos 2m ^ x 1) 



|2 



L3 



= — (6 5 sin 6m - 6 x 4 5 sin 4m + 15 x 2 5 sin 2m) 
32 v 

Hence <J> = m + e sin m 

e 2 e 3 3 „ . . _ . , 

+ - sin 2m + — x 7 (3 sm 3m - sin m) 

2 L 3 4 

e 4 e 5 1 

+ — x 4 (2 sin 4m - sin 2m) + yr x — x 

(5 4 sin 5m - 5 x 3* sin 3m + 10 sin m) + — . — 
(6 5 sin6m - 6 x 4 5 sin 4m + 15 x2 5 sin 2m) + . . 

Separating sin m, sin 2 m ...etc. 



x 



J- o 4 

e e 



<t> = m + (e - - + j£) sinm + {- - - + ^) 



sin 2m + ( 



3e 3 27e 



6 48' 
4 ,._6 



8 



i28 )sin3m + (|-^-)sin4m 



True Planets 123 

125e 5 
+ -rrr~ sin 5m + . . . (7) 

Next quantities contains powers of e or more 
hence they are very small and left out (e is very 
small because orbit is almost circular with very 
small eccentricity) 

Equation (1) can be also written as 

e sin O = <S>-m 

<I> - m 

or sin <l> = 

e 

From (7), this becomes 

2 e 4 e 3 e 5 

sin <D = (1 - — 4- — ) sin m + (- - — + ^) sin 2m 



2 „_4 / e 4e 5N 



e* 27c . m 

+ (3 T"W sin3m + 



125e 4 



15 



\ 



sin 4m H — — — - sin 5m + . . . . (8) 

384 

Now expansion of Sin 2 O, sin 3 O — are to 
be obtained 

Now in equation (6), take F (<I>) = sin 2 <!>, 

then F(m) = Sin 2m and F' (m) = 2 sin 2 m 

Hence equation (6) becomes - 

sin 2 <p = sin 2m + e sinm x 2 cos 2m 
e 2 d . 2 . . e 3 



+ Tr - — (sin m x 2cos 2m) + 



|_2 dm v J L 3 

12 e 4 d 3 



(sin m x 2 cos 2m) + 



.2 v J 1 4. j_3 

9 



dm 2 L 4 dm : 



224 



Siddhdnta Darpana 



♦4 



(sin m x 2 cos 2m) + 



e 5 d 4 



■_5 



L^dm' 



(sin m x 2 cos 2m) + ... 



In this, sin m x 2 cos 2m = sin 3m - sinm 

1 - cos 2m 



•_2 



- — (sin m X 2 cos 2m) = - — 
dm dm 



x 2 cos 2m 



\ 



dm 



(cos 2m - cos 2m) = 



dm 



cos 2m — 



1 + cos 4n0 



V 



= 2 sin 4m - 2 sin 2m 
d 2 



dm : 



(sin 3 m x 2 cos 2m) 



dm' 



[3 


sin m 


— 


sin 


3m 


\ 




4 







x 2 cos 2m 



f 3 sin m cos2m - sin 3m cos2m" 



dm J 



/ 



dm j 



dm' 



3 1 

- (sin 3m - sin m) - - (sin 5m + sin m) 

4 4 



— (3 sin 3m - 4sin m - sin 5m 



1 -y i 

= - (- 3 sin 3m + 4 sin m + 5 sin 5m) 

4 



•4 



dm' 



(sin m x 2 cos 2m) = 



dm' 



— (cos 4m — 4 cos 2m + 3) 2 cos 2m 
8 



-5 - (2 cos 4m . cos 2m - 4 x 2 . cos 2m 

dm 3 18 V 



+ 6 cos 2m) 



True Planets 



dm* 



- (cos 6m + cos 2m) 
8 v 



125 



4 6 

— (1 + cos 4m) + — cos2m 

8 8 



dm' 



- (cos 6m - 4 cos 4m + 7 cos 2m - 4) 
8 



= i (6 3 sin 6m - 4 4 sin 4m + 7 x 2 3 sin 2m) 
8 v 

so {sin2 O = sin 2m + e (sin 3m - sin m) 

+— (2 sin 4m - 2 sin 2m) 
e 3 1 

+ — - (25 sin 5m - 27 sin 3m + 4 sin m) 
[3 4 V 

e 4 1 
+ — . - (216 sin 6m - 256 sin 4m + 56 sin2m) + 

13 8 

e 3 2 7e 4 

= (- e + — ) sin m + (1 - e z + — ) sin 2m 

9e 3 ? 4e 4 25e 3 

+(e- — ) sin3m+ (e z - — ) sin4m + -^- sin5m + .... 



8 

In equation (6), now take F(0)= sin 3<£, then 
F(m) = sin 3m and F(m) = 3 cos 3 m, then it 
becomes - 

sin 3 O = sin 3m + e sin m x 3 cos 3m + 

2 



[2* dm 



d 2 

(sin m x 3 cos 3m) 



e 3 d 2 f 
+ -7-r . r | sin 3 m x 3 cos 3m + ...K 



[3 'da. 2 



4 



3e 



= {sin 3m + — (sin 4m - sin 2m) 

e 2 3 
+ — . - {(5 sin 5m - 6 sin 3m + sin m) 
3 4 v 



9 



226 



Siddhanta Darpana 



+ 



or, 



t 1 

6 ' 8 



(36 sin 6m - 48 sin 4m + 12 sin 2m) + ... 



sin 3 <J> = 



3e; 
8 



sinm 



3e 3e' 



(y- 4 



9e z 
) sin 2m + (1 — — — ) sin 3m 



3e 2 15e z 9e 3 

(— - 3e ) sin 4m H — - sin 5m + — — sin 6m + .... 

Similarly sin 4 <J> = sin 4 m + e sin m x 4 cos 4m 

e 2 d 
+ — . -r- (sin 2m X 4 cos4m) + — 
2 dm ' 



= Sin 4m + 2e (sin 5m - sin 3m) + — (6 sin 6m 
- 8 sin 4 m + 2 sin 2m) + 



2 2 

or sin 40 = e sin 2m - 2esin 3m + (l-4e ) sin 4m 



+ 2e sin 5m + 



Sin 5 <I> = Sin 5 m + — e (Sin 6m- Sin 4 m) + — 

5e 5e 

= — — - sin 4 m+ sin 5m + — sin 6 m+ — 
2 2 

Value of p can be known in terms of e by 
expanding with binomial theorem also (Taylor's 
theorem is not needed) 

_ VI + e - vT^~e 
P ~ Vl + e + VI - e 



1 - VT"^" 



1 2 /2 



e 

.6 



1 



1 e e e 

e ( T + ¥ + 16 + "") 



or P = 2 + T + ^ + - 



P 2 = 



/ 3 5\ 2 

e e e 



4 + 8 + 64 



True Planets 



12? 



3 fe e 3 e 5x 

e 3 3e 5 9e 7 ' 
= ~8~ + 32 + 128 + 



V e 4 . 5^ 

4 + 8 + 64 



'e 2 e 4 5e 6 
— + — + — 

4 8 64 



16 + 16 + 



/ 



.5 _ 



e 
16 + 16 



3 5\ 

£ £1 £1 

2 + ~8~ + 16 



e 
= 32 + 



Now equation (4) can be written as 



= m + 



sin 2m + 



3 5\ 

e e 

e ~ ~8~ + 192 



V 



3e 3 27e~ 



:\ 



sin m + 



8 



128 



\ 



/ 



■\ 



sin 3m + 



/ 



£l_ £ £! 

I 2 " 6 + 48 



e 

~3 



15 



/ 



125e 3 
sin 4m + — — — sin 5m + . . . . 

3 



+ 2 { ( f + y + i6 ) [ (1 ~T + ^ )sinm 



e e 2 e 5 

+ ( 2-T + 5 )sin2m 



+ ( 



3e 2 27e- 



4e- 



8 



118 



) sin 3m + (— - — - ) 



15 



125e :> 
sin 4m + sin 5m + .... ] 

1 e 2 e 4 5e 6 

2 7e 4 
sin m + (1 - e + — 



(-e + j) 



) sin 2m 



9e 4e 25 e 

+(e - — -) sin3m+(e 2 - — r-) sin4m H — — — sin5m +..] 



1 

+ - 
2 



'e 3 3/ 



32 



\ 



/ _ 



3e' 



8 



3e 3e 3 « 
sin m — (— - — - ) sin 2m 



128 



Siddhanta Darpan 



9e 2 



+ (1 T~ ) sin 3m 



(3e 5 15e 2 9e 2 

+ ~Y ~ ^ ) sin4m+ — sin 5m + — sin 6m +....J 



1 ~ 4 6 

l e e 



+ 4 ^16 + 16^ C sin2m ~ 2e sin3m + 

(1 + 4e 2 ) sin 4m + 2e sin 5m] 
1^. 5e 5e 

5 32 *~ T Sm 4m + sin 5m + Y sin 6m ^ 

Terms beyond e 6 and sin 6 m have been left 
out as they are neglible. Collecting the multiples 
of sin m, sin 2m —etc. 

e=m+(2e _^ + ^ )sinm +( ^_m + i^ 

4 96 ^4 24 192 

o- o , / 13e3 43e5 103e 4 451e 6 

sm 2m + (— - — - ) sin 3m + ( i^_ __ ^_ . 

12 64 v % 48Q ) 

. A 1097e 5 
Sm ~960~ Sin 5m (9) 

Actual equation for knowing heliocentric true 
position - 

Equation (9) is the main equation from which 
helocentric position of planets are calculated from 
their mean speeds and eccentricity of orbits. This 
is called manda karna in Indian system. For 
example, in case of Jupiter, e = 0.048254, hence e 2 
= 0.0023284, e 3 = 0.0001124, e 4 = 0.0000054, e 5 and 
higher powers are very small and can be neglected 
for calculation of 1" accuracy. 

For Jupiter - 

= m + (0.0 96508- 0.0000281) sin m 

+ (0.002 9 106 - 0.0000025) sin 2m 



True Planets J29 

+ 0.0001218 sin 3m + 0.0000058 sin 4m + - 
or 6 = m % + 0.0964799 sin m + 0.002 9081 sin 2m 
+ 0.0001218 sin 3m + 0.0000058 sin 4 m + — 
(10) 

If the sines are expressed in kala or vikala in 
Indian system, then the value of 6 will come in 
kala or vikala and this will be manda phala of 
guru from centre of sun. If they are expressed in 
fractions, the terms after m will be in radian. To 
convert them in kala or vikala, they are to be 
multiplied by 3437.75 or 206265. 

Equation for any planet can be obtained by 
putting its eccentricity e in equation (9) The 
eccntricities are given in end of this section. 

Helocentric distance - 

Manda karna (Heliocentric distance of planet) 

SP = a(l-e cos O) 

Putting F(*) = 1 - e cos 4>, F(m) = 1-e cosm, 
eiv? = 6 Sm m ' in e< l uation &)> ^ylor's series 



gives 



1-e cos <& = (l-e cos m) + e sin m — (l- 

dm 



cosm)+ 

L3 dm 



f — (Sin 2 m x e sin m 



1 3 • ^5 ( Sin m x e sin m) 



130 Siddhanta Darpana f 

1 

= 1-e cos m + — - — cos2 m + — cos 3m - — cos j 

4m + — cos 2m 

3 i 

e 2 3e 2 e 2 „ 2e 2 „ i 
= (1 + y ) - e (1 - — ) cos m - y (1 - — ) cos 2m : 



3e 3 



cos 3m + 



8 

Hence, radius (karna) 

e 2 3e 2 e 2 2e 2 x 

= a [ a + y ) - e ( x - -j- ) cos m - y ( a " X } 

3e 3 
cos 2m - — cos 3m] (11) 

Semi major axis (smallest+largest distance),/ 2 
of Jupiter a is 5202.8 hencee equation of its redius 
is 

5202.8 [ (1+0.0011642) - (0.048254 - 0.0000421) 
cosm 

- (0.0011642 - 0.0000018) cos 2m - 0.0000421 
cos 3m)] 

= 5202.8 (1.0011642 - 0.00482119 cos m - 
. 0.0011624 cos 2m - 0.0000421 cos 3m) 

= 5208.86-251.06 cosm - 6.05 cos 2m - 0.22 
cos 3m 

Semi major axis has been expressed as ratio 
of earth's mean distance from Sun which is taken 
as 1000 

Parameters of planetary orbit 

Constants for earth - a© = 1.4959787xl0 n 
metres, e is symbol for earth, a is semi major axis 
Time period of revolution T e = 3.1558150 x 10 7 sec. 



737 

True planets 

Mass m© = 5.976 x 10 24 kg, Moment M© = 
2 6 6 x 10 40 kg m 2 /sec 

Eccentricity e© = 0.0167 

Orbits of other planets 

Planet 



a in Peri- Mass Mome- inclinati- Eccentric- 
a© od (in m©) nt on of ity e 

Years (in m©) orbit 



Mercury 0.38 0.24 5.6 x 3*x V 0' 14" 0.2056 



71 



10 



10 



Venus 



Mars 



Jupiter 



Saturn 



Uranus 



0.72 0.62 8.1 x 7.0 x 3° 23' 39" 0.0068 



33 



10 



-1 



10 



-t 



1.52 1.88 1.1 x 1.3 x 1'51' 0" 0.0934 
37 10- 1 10" 1 

5.20 11.87 3.2 x 10 2 7.6xl0 2 1° 18' 21" 0.0484 
28 

9.53 29.46 9.5 x lO^.&riO 2 2'29"25" 0.0557 

89 

19.18 84.01 1.5 x lO^^xlO 1 0°46" 23" 0.0472 



Neptune 30.06 164.8 1.7X10 1 9.5X10 1 1°46' 28" 0.0086 
Pluto 39.44 247.6 2.0 x 1.2 x 17° 8' 38" 0.2486 

10' 3 lO' 2 

Conversion of Orbital distance to ecliptic 

distance Equation (10) gives the distance (angular) 
of planet in its orbit from its nicha (perihelion) or 
closest position. If the orbit of planet would have 
been in same plane as earth's orbit (or plane of 




A' P' 

Figure 4 - Inclination of orbit with elliptic 



132 Siddhdnta Darpana 

ecliptic), this would have been its distance in 
ecliptic also. But every planet's orbit is at an angle 
with ecliptic which is its parama sara (maximum 
distance from ecliptic). This inclination is given in 
the chart above. There is no inclination for earth's 
orbit (or sun) because it is measured from this orbit 
only. 

In fig 4, PC is orbital ellipse and CP' is the 
ecliptic. S is centre of sun and A is perihelion 
(nica) of the planet. P is true position. PP' is 
perpendicular on ecliptic, hence it passes through 
pole of ecliptic. Then ASP is orbital true anomaly 
(Kaksa spasta kendra) and SP is spasta karna. AA' 
is perpendicular on ecliptic and also passes through 
its pole. Distance A'P' along ecliptic is the ecliptic 
true anomaly (kranti vrttiya spasta kendra). 

For theoretical calculation, it is easier to find 
out relation between CA and CA' or CP and CP'. 
But in practice, we need to know only the minor 
correction to orbital distance to know ecliptic 
distance. 

This correction or difference between orbital 
distances from pata C (intersection point of orbit 
and ecliptic) is called parinati. 

Nica parinati = CA - CA' 

Planet parinati = PA-P'A 

PP' is instantaneous or istakalika sara, Z.PCP' 
is parama sara (equal to maximum angular distance 
from ecliptic), PC is distance from pata to graha 
or vipata graha. Z.PP'C is right angle, hence PCP' 
is a spherical right angle triangle. From Napier's 
laws - 



True Planets 133 

(1) Sin (90° - CP) - cos (PP') x cos CP' 

(2) Sin PP = cos (90 Q -PCP) x cos (90° -CP) 

(3) Tan PP' = Sin CP x tan PCP' 

(4) Tan CP' = cos (PCP') tan CP 

Sin (CP-CP) = Sin CP. cos CP - cos CP. 
sin CP -(12) 

From formula (3), 

tanPP 
sin CP ' = 



tanPCF 



¥ 

sin CP 
Formula (4), - = cos (PCP ) tan CP 

cos CP 

__' sin CP 

•. cos CP = 



cos (PCP) tanCP 



tanPP cos CP 

{ = ; x 



tanPCP cos PCP sin CP 

tanPP cos CP 
x 



sin PCP sin CP 
so, 

sin (CP - CP ) = sin CP -^ x ^g - cosCP 

sinPCF sinCP 

tanPP 



tanPCP 

tan PP x cos CP cosCP x tanPP 

sin PCP tanPCP 

tanPP x cosCP 



sinPCP 



[1 - cos PCP J 



__ sinPP cosCP 

: x 7 x vers sin PCP 

cosPP sinPCP 

From formula (2), — - t = sinCP 

sin PCP 



234 



Siddhanta Darpana 



Hence sin (CP-CF) 
sin CP x cosCP 

cos PP 



r 

x vers sin PCP 



Parama sara of all planets except Budha is 
less than 3.4° hence their istakalika sara will be 
still smaller. Hence Cos PP' s 1. Then 

Sin (CP-CP') = Sin CP cos CP x vers sin PCP' 
= 1/2 sin 2 CP. V sin PCP', or, sin (Parinati) = 1/2 
X versed sin of parama sara X Sin (2 x vipata 
graha) — (13) 

Equation (13) gives correction to find position 
of planet in kranti vrtta. 

Geocentric position 







Figure 5 - Geocentric position of planets 

To find the direction and distance of planet 
from earth, we have to know the position of earth 
itself. Position of earth also can be known from 
equation (9) like other planets. Position of Sun 
from earth is opposite to earth from sun direction 
i.e. 180° away. 

Sighra kendra is difference of ecliptic spasta 
kendra and position of sun from earth. 



TrU e Planets 135 

In Figure 5, S, E and J are positions of sun, 
ar th and Jupiter. ESS' is direction of Sun from 
6 a rth (both centres). S' is its position in ecliptic. 
c'Sj is sighra kendra of Jupiter. ZESJ = 180°-S'SJ 
and in AEJS, two sides ES, SJ and angle between 
them is known. Then EJ, L SEJ and ^EJS also can 
he known. From trigonometry 

^ «FJ-STE ST-SE SEJ + SJE 

tan 2 = SJ + SE tan 2 

Here L$ EJ+^SJE=ZS'SJ = sighra kendra 

SEJ-SJE SJ-SE sighra Kendra 

/.tan — — " S J + SE tan 2 

From this difference of angles Z.SEJ and ASJE 
can be known. Their sum (sighra kendra)) is 
already known. By adding these and dividing by 
2 we get ASEJ which is angle between Jupiter and 
Sun as seen from earth. This is called Inantara 
(Ina=Sun). 

Distance of Jupiter from Earth JE is sighra 

karna. 

TE JS fcI . 
Lz. - - — : by sm ratios 

Sin L E S J sin L S E J 

But sin ^ISEJ = Sin (180°-^SEJ) * ^JSS' = Sin (Sighra 



kendra) 




Figure 6 - Sighra Kendra in ecliptic 



236 Siddhdnta Darpana 

Hence, Sighra karna JE = Sin of SIghra kendra x manda 

karna x - — _ — - —(14) 

sin (inantara) ■ ■ ■ .' 

In fig 6, XX' is ecliptic plane which contains 
earth's orbit EYZ. Orbit of Jupiter is CJC which 
cuts ecliptic on C and C. C is north pata and C 
south pata. S,E and J are true positions of Sun, 
earth and jupiter.*- JJ' is perpendicular on ecliptic 
plane. V is point of vernal equinox (north pata of 
ecliptic and equator planes). ZJSJ'is heliocentric 
inclination of Jupiter, Z.VS]' is longitude of planet 
(angle in ecliptic plane between vernal equinox and 
planet - seen from sun. JEJ' and V'EJ' are geocentric 
inclination and longitude of jupiter. SVMEV. 
^V£E is heliocentric longitude of earth, hence 
ZVSE + 180° is geocentric longitude of sun. 

(A SEJ' is same as A SEJ of figure 5) SIghra 
karna and inantara in equator are EJ' and Z.SEJ'. 
True sighra karna and inantara are EJ and Z.SEJ; 

EJ = cos^JEJ- ~< 15 > 

L J E J' is very small, hence is cosine is almost 
1. 

This is only a rough outline of calculation of 
palnetary positions in modern astronomy. 

There are prturbations in positions of earth 
due to effect of moon, jupiter and venus (others 
negligible) Similarly prerturbations occur in other 
planets also for which corrections are necessary. 
There is slight change in eccentricity and positions 
of pata also which cause other corrections. The 
corrections to the orbit of moon are more important 
because it has largest effect on earth's tides, 
climates, calender and eclipse etc. 



frue Planets 137 

(3) Tables of Sun - 

Precession of equinoxes - According to New- 
comb, rate of general precession in longitude per 
tropical year of 365.2422 days is 50". 2564+0". 02223 
(t/100) + 0".0000026 (t/100) 2 

where t is measured in tropical years from 
1900.0 AD. 

Annual rates of precession per sidereal year 
f 365.25636 days is 50".258 35 + 0". 02223 (t/100) 
+ 0". 00000026 (t/100) 2 

In Julian year of 365.25 days, precession is 
50."25747+0".02223 (t/100) + 0".0000026 (t/100) 2 

In Indian system, initial point from which 
longitude is measured is a fixed point of ecliptic 
with respect to stars. In modern astronomy, it is 
the point of vernal equinox. Distance from fixed 
initial point of vernal equinox point is called 
Ayanamsa. 

To fix initial point accurately, star spica 
(avirginis) has been assigned a nirayana (from fixed 
point) longitude of 180°. Since the star also has 
some small motion, its longitude of epoch time is 
taken when fixed point and vernal equinox point 
were together with sun on it. 

This epoch of 0° Ayanamsa (0° sun also) was 
on 285 AD, March 22, 17h 48m E.T. or 21h 27m 
1ST. That was beginning of saka era 207, Samvat 
era 342 and Kaliyuga era 3386. Julian day on March 
22 noon was 1825325 and kali elapsed days at 
midnight was 1236770. The day was Sunday. Mean 
su n (both tropical and sidereal) was 0°0'0" and 
(Mean moon - Mean sun) = 351 '.67. Thus it was 
also a new moon day. In Besselian fictitious year, 
epoch was 



138 



Siddhdnta Darpana 



285 



79.994 
360 



285.2222 A.D. 



The epoch is 1614.7778 years before 1900.0 
AD. From 0° Ayanamsa of this epoch to ayanamsa 
at 1900 AD, January 0.813 i.e. 19h 31m ET is 
22°27'43".51. Thus Ayanamsa from 1900.0 AD is 



' t x 



100 



A = 22°27'43". 51+50". 2564t+l". 1115 
3 



't^ 2 



100 

\ / 



+0".0001 



This is formula in tropical years. In sidereal 
years, it is 



A = 22°27'43."40+50".25835t+l".1115 



't* 2 



/ t v3- 



100 



+ cr.oooi 



100 



\ 



t 



In Julian year, formula is 

A = 22°27 / 43". 40+50". 2575t+l". 1115 

3 



't* 2 



100 



100 

\ / 



+ 0".0001 



Daily rate of precession in 1900 AD = 
0". 137597 

If time is taken from 285AD epoch, formula 
in tropical years is 

A=49".8981t+l".1073 (t.100) 2 + 0.0001 (t/100) 3 

+ — 

There are similar formula for sideral and Julian 
years. 

Position of star Spica of 180* long in 285 AD - 

In 1950.0 AD its position was 



139 
True Planets 

R.A. - 200° 38' 19". 6, Declination = -10'54'3".4 
Annual proper motion A« = -0".039 A<5 = 

_0".033 

Tropical longitude = 203° 8'36"3 latitude -= 

-2°3'2".8 

Sidereal long = 179°.58'59".7 (Ayanamsa 

23°9'36".6) 

Annual proper motion in ecliptic system is 

*AA = - 0".0232, A0 = -0". 0449 
Due to slow motion of plane of ecliptic, 
longitudes and latitudes of fixed stars undergo 
changes. Annual rates are as follows - 

AA=7rcos(A-n)tA n £ A /3 = - * sin (A - II) 

In 1950 AD I* 285 AD 

[1 = 0".4708 0^.4824 

A=203° .4', p = -2° 3', A = 180° 0' ,fi = - 1'56' 

[1 = 174°24' (Trop) 159° 12' 
Hence AA = - 0". 0147 - 0". 0151 

A p = -0" . 2283 -o" • 1713 

Average value of AA= -0".0149, A0 = - 0".1998 
Proper motion - 0".0232, - 0".00449 
Total anual variation AA= -0".0381, A/3 = 
-0".2447 

In 1665 years (1950-285 AD), total variation in 
longitude is - 63". 4, in latitude - 6'47".4. Then 
nirayana longitude in 285.22 AD is 180°0'3".l and 
latitude is - 1°56'15".4 Thus at epoch, ^its nirayana 
longitude was 180° approx. 



'?■# 



40 Siddhanta Darpana 

Obliquity of Ecliptic to the equator - 

E = 23°27'8".26-46". 845T - 0".0060T 2 
+0".001837T 3 

where T = Julian century of 36525 days from 
1900.0AD E.T. 

Rate of variation per century is 

de 

— = - 46". 845 - 0".012 T + 0". 00549 T 2 

When T 3 term has appreciable value, century 
figures need some correction. Then putting T = 

Tc+ I5o 

(Tc = completed centuries, t = extra years) 

e = 23'27'8".26-46" 8457 Tc - 0".006T 2 + 
0.00183TC 3 

+ (-0".00651 Tc + 0".00549 T c 2 ) x — 

; 100 

Mean Longitude of Sun - (L) - Epoch is 1900 
AD, Jan 0.0 ET. i.e. 0h0'4".4 universal time, T = 
Julian centuries of 3 6525 ephemeris days from 
epoch. According to Newcomb, sun's mean tropical 
longitude, freed from aberrations is 

L=279'12 / 13 // .88 + 129602768 7 '. 13 T+l".089 T 2 
Motion in a century of 36525 ephemeris day 
is 12960 2768/'13 = 360° x 100+27".6813 x 100 
= + 46'8 /7 .13 

Daily motion is 0°59 / 8 ,, .3304074 

If Tc is completed century, t = remaining 
years, d = extra days, 



True Planets 142 

L = 277°!2'13".88 + (46'8".13» Tc + (59'8".330) 
d + 1".089 Tc 2 + 2.178 Tc x ^ 

Sidereal or Nirayana Mean Sun (L') is 

V = 256°44'30".48 + 129597742". 38T-0".0225T 2 

4)."0001T 3 

Motion in a century = 360° x 100-22".5762 x 
100 = -37'37".62 

Daily motion = 35488". 192 80988 = 
0°59'8".1928098 

Sun's Perigee (=11) and Mean anomaly (=g)) 

Trop II = 281°13'14".92 + 6189".03 T+1".63T 2 
+ 0".012T 3 

Sid IF = 258°45'31".52 + 1163". 28 T + 0."52 
T 2 + 0".012T 3 

Motion of IF per century = 19'23".28, per year 
= 11". 63, per day = 0".0318 

Mean anomaly of the earth or the sun 

= g = L - II or L' - IF 

g = 357°58'58".96 + 129596579." 10 T-0".541T 2 - 
0".012T 3 

Daily motion = 0°. 9856002670 = 3548". 160961 

Hence the period = 365.2596413 ephemeris 

days. 

Mean anomaly M in days is obtained by 
dividing g by daily motion and adding a constant 
of 5.37018 days 

M = 3.12376 + 36525 T - 0.0001525T 2 
" °- 0000034 1 ; * 



242 Siddhdnta Darpana 

36525 days = Period x 100 - 0.96413 days. 

Mean Elongation of the Moon in days 

Brown's Moon 

= 263°50'45".48+1732564379".31T - 4".08T 2 +' 

0".0068T 3 

Newcomb's sun = 279° 12'13".88 + 
129602768". 13T + 1".089T 2 

D=Moon - sun 
= 344°38'31".60+1602961611".18T - 5".169T 2 + 
0".0068 T 3 

Daily motion of D = 43886" . 697089 
Period = 29.53058867 days 
Converting into days 

D = 28.27079 +. (period x 1236+25.192399) T 
- 0.0001178T 2 + 0.000000155T 3 

Venus and Sun - Mean Tropical Venus is 

341°57'57".49 + 210669162". 88T + 1M148T 2 

V = Venus - Sun 

= 62 °45'43". 61 +81066394. "75T + 0".0258T 2 

Daily motion of V = 2219". 4769, period = 
583.921373 days. Converting into days 

V = 101.8004 + (period x 62 + 321.87487) T + 
0.0000116T 2 

Sun and Jupiter - Mean Tropical Jupiter is 

238 o 0'27".69 + 10930687. 15T + 1".205T 2 

J = Sun - Jupiter = 41° 11'46".19 + 
118672080".98T - 0M16 T 2 

Daily motion of J = 3249". 064503 



Period = 398.884048 days 

Convereting into days, we get 

j = 45.6458 + (Period x 91 + 226.55163) T 
0.000036T 2 
Nodes of Moon - Tropical longitude of the 

node is 

Q = 259 12'35".11-6926911".23T+7".48T 2 

+ 0".008T 3 

- Q = 100°47'24".89+6926911". 23T-7",48T 2 

_ 0".008T 3 

Daily motion = 190". 63412, Period = 
6798.36327 days converting into days expression 
for - Q and adding a constant of 0.818 days, No. 
of days N since tropical longitude of moon's mean 
node was zero is 

N = 1904.177+ (period x 5 +■ 2533.1835) T - 
0.003924T 2 - 0.000042T 3 

Julian day Number : 

Pope Gregory introduced in 1582 AD year of 
365.2425 days by omitting 10 days (Oct. 5 to Get. 
14)) from calender. Before that, there was leap year 
in every 4 years. In Gregorian calender, 97 leap 
years come in 400 years. Years divisible by 4 or 
centuries by 400 are leap yeears of 366 days. Normal 
year is of 365 days. 

Julian days are numbered serially from Jan 1, 
47 13 B.C., Monday at Greenwitch mean noon. 

Besselian Fictitious year begins when the 
topical mean sun is 280°0'20".5 or the same 
unaffected by aberrations is 280 °0'0". Notation like 
^00.0 AD. is used for this year. 



144 Siddhdnta Darpatut 

Let K = time from beginning of Besselian year 
upto beginning of calender year i.e. Jan 0, oh E.T. 
for common year or Jan 1, Oh E.T. for leap year. 

Day from beginning of ficitious year = Day of, 
year + K 

K = - 0°48'6".6+129 602768M3T + 1".089 T 2 

Daily motion = 3548" 3304074 

Period of length of Tropical solar year = 
365.24219878 days 

K in days = - 0.8135 + (perod x 100 + 0.780122) 
T + 0.000307T 2 

Inequalities of long period in mean longitude 

<5L=+6".40 sin (231°. 19+20°. 20T)+(1".882- 
0".016T) x sin (57°24+150\27T)+0".266 sin 
(31°. 8+119 °.0T)+0".202 sin (315\6+893°.3T) 

First term has a period of 1782.2 years (century 
variation of 20°. 2) i.e. 1° in 3548 days. 

<5L = + 6".40 sin [(AD year - 755.5)x0.202)] 

Equation of Centre: e = eccentricity of orbit/ 
g = mean anomaly (written as m in derivation of 
formula). 

Equation of centre 

e 3 5 2 He 4 13 % 

(2e - -) sin g + (-e 2 - — ) sin 2g + — e 3 

. „ 103e 4 
sin 3g + -^psin4g 

Here, e = 0.016,751, 04-0.000,041,80T- 
0.000,000,126T 2 = 0.016, 75104-0.000,041,80 (T+.00 
301T 2 ) 

Multiplying by 206264.8 we get 
e = 3455". 150 - 8".621 (T+0.003T 2 ) 



True Planets 145 

So equation of centre = + 6910 / .057 Sin g + 
72'.338 sin 2 g + 1".054 sin 3 g + 0".018 sin 4g 

•- 17".240 (T + 0.003T 2 ) sin g - 0".361T sin2g. 

perturbations to Sun - 

Action of Moon - Longitude of sun (or earth 
in opposite direction) is the longitude from centre 
of mass of earth and the moon. This is called 
geometric longitude. The origin is to be transferred 
to centre of earth. 

Radius of earth is taken as unity, f]' an d 17 
are horizontal parallaxes in seconds of arc of moon 
and sun respectively, /}' are /? are their latitudes. 
Distance of mass centre from earth centre in 
direction of moon 

206265 _ 2506.3 
" 82.30 ft' I! ' 

(Ratio of earth mass to moon mass is 81.30 
adopted in 1968) 

AL = 2506.3 x ^ Cos£' sin p-O) 
P = 2506.3 x Jl Sin p 

Substituting numerical values - 

AL = + 6". 44 sin D - 0".42 sin (D-g') 

Sun's latitude /J = + 0",58 sin U, or + 6". 44 
sin fi' or roughly 0.11 x moon's latitude in seconds 

U = Mean moon - lunar node 

A log R = + 0.0000134 Cos D. 

Action of other planets 

Action is calculated in terms of Q = difference 
m heliocentric latitudes of the planet and earth. 



146 Siddhanta Darpant^ 

Due to elliptical shape the deviation due to planets 
also depends on g and (W'-W) where 

g = mean anonaly of earth (i.e. of sun) j 

W = Longitudes of the planet's perihelion, | 
W = perihelion of earth J 

T is in 100 years from 1850 AD then | 

K' = W'-W = 29 •5'55 ,, -18 / 40 ,/ T (Venus) 

K" = W"-W = 232°56'11" + 7'18"T (Mars) 

K "' > W w -W = 271°33'16" - 6'33"T (Jupiter); 

Century variations of these quantities are very 
small, and they dm be 'considered as constants fbfcj 
1000 years or more 

g = hehiocentric lat. of earth - W 
g' = hel. long, of the planet - W (e.g. for :\ 
venus)) 
, = planet - k'-W 

= (Planet-earth) + (earth - W) - K' 

= Q + g - K' 

Perturbations due to venus (Approx New- 
comb formula) 

Pert = + 4.84 sin Q - 5.53 sin 2 Q-0.67 sin3Q 

- 0.21 sin 4Q - 0.12 sin (2Q+g) - 2.50 ski 
(g+12°-2Q) 

- 1.56 sin (g+12°-3Q) + 0.14 sin (g+12°-4Q) 

- 1.02 sin (2 g + 40°-3Q) - 0.15 sin (2 g+40°-4Q| 
+0.12 sin (2g+40 # -5Q) - 0.15 sin (3g+56°-5C$ 
Corresponding formula given by Le-Verrier fej| 



True ^^ s U7 

Pert = + 4.91 sin Q-5.61 sin 2 Q-0.67 sin 3Q 
21 sin 4 Q - 2.52 sin (g-2Q + W-90°) - 1.58 sin 
(g-3Q+w-9(T) 

For first approximation, calculation is based 
on Q only, then for M = g+5\29. it is calculated 

Perturbations due to Jupiter - Newcomb 
formula is 

Pert = + 7.21 sin (Q-l°5')-2.73 sin (2 Q - 0°15') 

- 0.16 sin (3 Q + 4'5r) + 2.60 sin (Q+g-84'46') 

- 1.61 sin (2 Q + g - 22°. 6) - 0.56 sin (3 Q 
+ g + 87°2) 

- 0.16 sin (g-Q+20'.l) - 0.21 sin (3 Q + 2 g 
+ 77°) 

First three terms according to Le verrier are 

+ 7/20 sin (Q-l°5') - 2.73 sin (2 Q-18') - 0.16 
sin (3Q + 5°) 

These terms are tabulated for Q, then for Q 
and M. 

Perturbations due to Mars - Newcomb for- 
mula is 

Pert = + 2.04 sin (2Q+15') + 0.27 sin (Q~0°.6) 

- 1.77 sin (2 Q+g-36°16') - 0.58 sin (4Q+2g+84°) 

- 0.50 sin (4 Q + g - 47°) - 0.43 sin (3 Q + 
g - 47\7) 

Aberrations - Correction in longitude due to 
aberration of light in earth's atmosphere is 

- 20".50 - 0".34 cos g 

Nutation 

Tropical longitude is calculated from mean 
e quinox of the date. Correction due to nutation is 
*° be made in tropical longitude, but, not necessary 
w nirayana longitudes. 



I 



248 Siddhdnta Darpan 

Solar nutation = - I". 27 sin 2L+0M3 sin g 
- 0.05 sin (3L+79 ) 

Lunar nutation = - 17". 23 sin Q + 0".21 sm 

2Q i 

Principal term of the lunar nutation is slowly) 
increasing at the rate of 0".17 per thousand years3 

In the obliquity . of ecliptic, 

Solar nutation = + 0."55 cos2L + 0.02 cos 1 
(3L+79°) 

Lunar nutation = + 9".21 cosQ - 0".09 cos 2Qj 

Here L and Q are the tropical mean longitud 
of the Sun and the lunar node respectively. 

Radius Vector 

Radius vector is expressed in terms of meanl 
distance of earth from sun. Mean distance isj 
expressed by Gauss formula based on Keplar's thircji 
law 

aV = k 2 (1+m) 

-where k is Gaussian gravitational constant * 
3548". 187607 

m = mass of earth and moon, taking su 
mass as unity 

n = observed sidereal mean daily motion o 
earth. 

a = mean distance from sun to mass centr 
of earth and moon. 

Value of k is based on sidereal period 
365.256898 days of earth considered as partid 






True Planets 14$ 

without mass or of 365.256344 days with adopted 
value of mass. 

With Newcomb's value of m = 1 + 329390 and 
n = 3548 ,, .19282 / we get log a = 0.000,000,013. 

Long term effect of attraction of inner planets 
is equivalent to an increase in mass of sun, to 
balance it, radius vector a increases. Observed daily 
motion n remains constant. 

Elliptic term of radius vector is (equation 11) 

R = a n + y -. ( 3,3) cosg-&-\ e * )coslg 



3 3 1 , 

- ge cos 3g - -e* cos 4g ] 

J. 2 q 

and log R = log* + log (1 + - ) - M [ (L _ 1^ 
+ (e - |e 3 ) cos g + (|e 2 - ge 4 ) cos 2g 

_l 17 3 . 71- 4 

+ ^je cos3g + — e 4 cos4g] 

where M is the modulus of common logarithm 
s 0.434294. & 

Taking value of e for 1900, 

nn^ R = L000 ' 140 ' 5 " °- 016 ' 7 ®> 2 cos e - 
0.000,140,3 cos 2 g 8 

nn ' 0.000,001, 8 cos 3 g - 0.000,000,7 T + 
0000,04,18 (T+0.003T 2 ) cos g + 0.000,000, 71 Cos 
8 

n nn„ L ° 8 R = °- 000 ' 03 °>6 - 0.007,274, 1 cos g - 
UU0C 1,091, 4 cos g - 0.000,0015 cos 3 g - 0.000,000,15 

i, + 0.000,018,14 (T + 0.003T 2 ) cos g + 0.000,000,46 
1 cos 2 g. 



• 



ISO Siddhanta Darparut 

Terms free of T are value for 1900 A.D. Terms 
with T are secular variation. 

Effect of planets on radius vector - 

Due to venus = - 1".12 cos Q + 3".25 cos 
(2Q+7') + 0".50 cos (3 Q - 1°5) + 0'M8 cos (4Q-2\4) 

+ 0".08 cos (5Q-3 ) 1 

Log = - 2".36 cos Q + 6.84 cos (2Q+7') + 1.0S 

cos (3Q-1°5) + 0".38 cos (4Q-2\4) + 0.16 cos (5 

Q-3') 

Jupiter = + 3".36 cos (Q-l°6') - 1 ,# .91 cos 

(2Q--13') - -0.13 cos (3 Q + 4°3r) 

Log = 7.07 cos (Q-l°6') - 4.03 cos (2Q-130-0.28 
cos (3Q+4°2r) 

Due to Moon = + 6". 35 cos D or log = + 
13.36 cos D. 

Sun's semi diameter and horizontal parallax - 

At unit distance, apparent semi diameter ol 
sun = 961 ".18 and horizontal parallax = 8". 794. Atl 
any distance, Semi - diamter = 961M8 /R ! 
16'1 / M8+16'M0 cos g+0".27 cos 2 g 

Parallax = 8 ,, 794/R = 8".79+0".15 cos g. 

For calculation of eclipse, allowance of 1".5 
is made for irradiation. Then true semi diamet 
at unit distance is 959". 63 

Reduction of Rt ascension and declination 

A = tropical longitude of the sun, a = righj 
ascension, 



152 
True Planets 

d = declination, e = true obliquity of ecliptic 

t0 equator, 

, d<3 
Sin d = Sin A sin a and — = sin a 

tan a = tan A. cos e 

1 * 

or a = A- (tan 2 § sin2A- - tan 4 2 sin 4 A 

+ 1/3 tan 6 I sin 6 A ) 

^ = - i sin 2 a tan e = - 0.2168 sin 2 a 
de 2 

Sidereal Time - Sidereal time at any instant 
is defined to be west hour angle of the First point 
of Aries (Vernal equinoctical point) from the upper 
meridian of the place. 

Sidereal time at mean noon (i.e. 12h local 
mean time) on any day is the right ascension of 
the fictitions mean sun, which is defined to be the 
tropical mean sun at moment as affected by mean 
aberration. 

At mean midnight, sidereal time is 12h (i.e. 
180°) + R.A. of fictitions mean sun for the moment. 
Sidereal time at Greenwich mean midnight = 
6h6 m 47 s ,558 + 8640184 s .542 T+0 5 .0929T 2 

where T is Julian centuries of 36525 days from 
1900 AD, Jan 0, h or EX 

Motion in a century = 100 d + h 3 m4 s .542 
Motion in a day = 3 m 56 s .5553605 
Equation of Time = Local apparent time - 
Local mean time 



* 






152 Siddhanta Darpana 

Local apparent noon = 12 h L.M.T - equation 
of time (E) 

At h E.T., Equation of time = Apparent 
sidereal time - Apparent R.A. of sun. 

E = R.A. of mean sun - R.A. of true sun 

Both are affected by aberration and nutation. 
True sun is also affected by perturbation. Omitting 
aberration and nutation from both sides, only 
perturbation A remains in true sun. 

True Sun = L + equation of centre 
E = L - (L+Eqn of c) + tan 2 ~ s * n 2A 

1 e i E 

- — tan 4 ^sin 4 A+ — tan 6 '-* sin 6 A 

- effect of perturbation in longitude 

Equation of centre in seconds of time is 

+ (460.67 - 1.149T) sin g + 4.82 sin 2g+0.07 
sin 3g 

Value in arc is, tan 2 | = 0.0430836-0.0000491 



In seconds of time, tan 2 | = 592.44r-0.675 T 

So equation of time (in seconds of time) is 

= - (460.67-1.149 T) sin g - 4.82 sin 2g - 0.07 
sin 3g + (592.44 - 0.675 T) sin 2A - 12.76 sin 4A + 

0.36 sin 6 A - — perturbation in longitude. 

15 



m 



A 



i 



■-■.'I 



■$ 



True Planets 153 

4, Equation for other planets 

Basic constants of Mercury 

Mean longitude, L for 3200 BC, Jan 0.5 epoch 

is 

L = 49°. 677936 + 538106654".8 T-1".084T 2 

L for 1900 AD epoch is 173°. 303523 (51 
centuries - 13 days) 

Mean anomaly, g for 3200 BC is 

g = 53°. 107661 + 538101055.04T - 0".024T 2 

g for 1900 AD is 98°. 169610 

Argument of latitude, U is for 3200 BC 

U = 62'. 977228 + 538102388".05T-0".458 T 2 

U for 1900 AD is 136 '.609863 

Constants for venus - 

Mean longitude L for 3200 BC (-51 centuries 
+ 13 days) 

= 285\18561+210669162".88T+1".1148T 2 

L for 1900 AD is 341°. 97032 

Mean anomaly g for 3200 BC is 

g = 223°. 83111+210664093.95 T + 4".63 T 2 

g for 1900 AD = 214°. 34622 

Argument of latitude U for 3200 B.C. is 

U = 252°.31206+210665923".42T+0".3612T 2 

U for 1900 AD is 266.59425 

Constants for Mars - for 3200 BC 

L=33°.370172+68910117".19 T - 1".1184T 2 

„ g = 152° 99708 + 68903493. 19T-0".651T 2 - 
0'\0192T 3 



154 Siddhdnta Darpana 

U = 23°. 923117 + 68907340."7 T - 1M234T 2 - 
00".00192T 3 

For 1900 AD epoch constants are 

L = 292° 416147, g = 318°. 387964 

U =242°. 918470° 

Constants for Jupiter - For 1900 AD are 

L = 238°.0496+10930687".148T+l".20486T 2 - 
0".005936T 3 

U = 138°.60587+10927049".24T + 0".06314T 2 + 
0".024704T 3 

g = 225°.32833+10924891".286T+2".59772T 2 + 
0".06314T 3 

Long period inequality in longitude L is 

E = (1186". 618572 - 0". 0347004 t + 
0".000033372t 2 ) SinC - 12".013596 sin 2C 

where t is number of years from 1800 A.D. 

C= 95°. 8814+0°. 38633184 t + 0°. 000035 It 2 

Constants for Saturn - for 320Q BC are 

L = 147°.9623+4404635".581T-1".16835T 2 
-0".021T 3 

g = 156°.74269+4397585".284T-1".80655T 2 

- 0"0376T 3 

U = 79°. 704558+4401492". 0785T-1".7162T 2 

- 0".0019T 3 

Constants for 1900 AD are 

L = 260°. 46036, g = 172°. 74219, U = 152°.43062 

Notes : (1) Newcomb's formula has been 
corrected by Rossi for Mars. There are other 
perturbations for planets also which have not been 
written. 



True Planets 155 

(2) Moon's motion in detail will be discussed 
in the next chapter. 

(3) From these constants, equations of centre 
and radius vector can be obtained. 

(4) Value of eccentricity also changes. For the 
present century, values given in chart can be taken 
as constants. 

(5) From these equations, constants are tabu- 
lated for centuries. Then by ratio, they are fixed 
for specific years. 

(6) Equations of centre and radius vector give 
true positions from the constants of year. 

(7) Longitudes and latitudes are reduced to 
ediptic. 

(8) Heliocentric longitude and latitude are 
converted to geocentric values. 

Let S he longitude of Sun, R its radius vector 

H is heliocentric longitude of planet, b its 
latitude 

r is radius vector from sun of planet, 
x is geocentric longitude, y is latitude 

r cos b sin (H - S ) _ 

then tan P = — . V m — "T , x = S + P 

R + r cos b cos (H - s) 

r sin b sin P 
tany = R + rcosbsin(H-S) 

5. References - (1) Any text book on modern 
coordinate geometry, Trigonometry can be referred 
for these formula. 

(2) Dynamics of planetary motion has been 
explained in Dynamics of Rigid bodies by A.G. 
Webster. 



156 Siddhanta Darpana 

(3) Derivation of formulas can be referred in 
books on spherical trigonometry. Inportant books 
are (1) Spherical Trigonometry by Gorakh Prasad, 
Pothishala, Allahabad - 2, (2) A hand book of 
Practical Astronomy by R.V. Vaidya, Payal 
Prakashan, Nagpur (3) Astronomy by G.V. 
Ramachandran, Tiruchirpalli (4) Practical 
Astronomy by Schroeder, published by Werner 
Lauries, London (5) Astronomy by R.H. Baker - D 
van Nostrand, East West Edition (6) Celestial 
Dynamics . by W. Smart - Longman, Green (4) 
Astronomical charts were published by Simon 
Newcomb in 1899 and 1906 but they are out of 
print. Nautical Almancs published by govt, of 
India, specially 1st edition of 1958 can be referred. 
Tables of sun have been published by Sri N.C. 
Lahari from Calcutta in 1993 (revised edition.) 

Translation of the text 

Verse 1 - Scope and definition - The position 
in which graha in seen from earth is to be found 
by calculation. This process is called Sphutikarana 
of graha or making is sphuta. The graha give 
results according to the position they are seen. 
Hence method for making a graha sphuta is being 
explained. 

Verses* 2-5 - Reasons of planetary motion - 

Celestial sphere containing graha and naksatra 
revolves around earth once in a day from east to 
west due to attraction of a wind (Pavana) named 
Pravaha rotating round earth. This is called daily 
motion. 

Graha move in opposite direction from west 
to east compared to stars (or naksatras) with slow 
speed according to their own enregy. This is called 
natural speed of a planet (svabhavika gati). There 



True Planets ^7 

are deviations from this average speed of planets 
under influence of ucca (sighra and manda). 

(According to Surya siddhanta) Invisible forms 
of kala like sighrocca, mandocca and pata residing 
in celestial sphere are reasons of planetary motion. 

Notes (1) Daily moiton is due to rotation of 
earth around its axis. Earth appears fixed to us 
and stars rotate in opposite relative motion. Reason 
of earth's rotation is due to initial conditions of its 
formation, now it continues due to inertia. That 
inertia is assumed to be 'Pravaha', an imaginary 
force in vacuum. This is similar to assumption of 
ether for propogation of light in vacuum. 

(2) True position of a planet is closer to 
mandocca (farthest point in elliptical orbit) com- 
pared to its mean position. Due to that reason 
attraction by mondocca in seen. Similar is case with 
sighrocca. 

(3) Pata is point of intersection of planetary 
orbit with ecliptic due to its inclination. Hence, 
pata appears to repulse a graha away from ecliptic. 

Verses 6-9 - Nature of motion - Mean sun 
moves around earth between naksatra and orbit of 
grahas. Other planets like mars are in orbit round 
mean sun and along with it, they also revolve 
round earth. Hence (mean) sun is called attractor 
of all. From dainika gati of Mafigala, Brhaspati and 
sani - dainika gati of ravi is more and they are 
attracted by ravi. Hence ravi is called sighrocca of 
these planets. 

Compared to Budha and sukra, speed of ravi 
is slower and it always remains between them. 
Hence Budha and Sukra are called their sighrocca. 



158 Siddhanta Darpana 

Notes (1) Outer planets are almost in same 
direction from earth as from sun. Minor correction 
is due to position of sun from earth. 

(2) Inner planets are within a small distance 
from sun which is their average position. First 
correction for their true position, is due to their 
own motion. Hence they are own sighrocca. 

Verses 10-16 - Slow, fast and reverse motion 

- Planets in successively farther orbits from sun 
are - budha, sukra, mangala, brhaspati and sani. 
Hence their angular speed appears progressively 
slower from earth (if linear speed in orbit is 
assumed to be same). 

Like ravi, moon also is rotating round the 
earth, but from very close distance. Hence angular 
speed of moon is largest, though its linear speed 
(in yojanas etc) is small. 

Budha and sukra are close to ravi, compared 
to earth. Therefore, they are seen with ravi after 
12 rasi (full rotation), as well as, after 6 rasi (half 
rotation). 

Mangala, brhaspati and sani are farther from 
ravi - compared to earth. Hence, they appear 
together with ravi at 12 rasi * difference and in 
opposite direection at 6 rasi difference. 

When earth is in one direction of ravi, and 
star planets (tara graha) mangala, budha, guru, 
sukra and sani are in opposite direction - then the 
graha appears to move in forward direction (margi 
gati) 

If a tara graha and earth are in the same 
direction of ravi, then the graha appears to move 
in reverse direction (vakri gati) due to difference 
between mean and sighra speeds. (Figures 7, 8) 



159 



True Planets 

Notes (1) Explanation of forward and reverse 
speeds - Figure 7 indicates relative speeds of earth 

0* Vernal equinox 




180' 
Figure 7 - Foreward and reverse speeds of inner planet (Budha) 

■ * 

Vernal equinox 




270" 



180' 



Figure B - Foreward and reverse speeds of outer planet (Mars) 

and an inner planet Budha. Figure 8 compares 
earth's motion with an outer planet mars. Numbers 

1, 2 8 indicate successive simultaneous positions 

°f the planets in their orbits. Ecliptic is a much 
bigger circle whose 0° stars' from vernal equinox. 
AU the circles are in same plane and the movements 
ar e in positive direction (anticlock wise). Line of 



:m 



■y 









260 Siddhanta Darpana 

sight from position 1 of earth to simultaneous 
position of planet 1 on a point on ecliptic is marked 
1. Point 1, 2 — 8 on ecliptic are apparent directions 
of the planets seen in ecliptic after regular intervals. 
Speeds in inner orbits are faster. 

Figure 7 shows that 1,2,3 positions indicate 
forward movement of Budha. After point 3, budha 
comes on same side of sun as earth and moves 
backwards in positions 4 and 5. Between 5 and 6, 
it is almost stationary before moving forward again 
to positions 7 and 8. 

Similarly, positions 1 to 5 in figure 8 indicate 
forward motion of mars. At 6, it moves back wards 
when earth and mars both are on same side of I 
sun. Actually ecliptic circle is at almost infinite 
distance and position 7 also is in backward direction 
from 6. But due to small construction of ecliptic it 
looks forward. From position 8, planet again moves 
forward. 

(2) Fast and slow speeds - Angular speed is 
arc length divided by radius in unit time. Hence 
for same arc length in unit time, angular speed 
will be less for large radius. Thus farther planets 
will look slower. In siddhanta text, this was 
considered only reason for slow speed. But linear 
speed also becomes slower as explained by Keplar's 
law (to balance lesser gravitational pull). 

Verses 17-18 - Five tara graha revole round 
the sun at constant distance in east direction. They 
are attracted by their mandocca and sighrocca ravi. 
They always* remain in bha-cakra (ecliptic circle). 

Earth is in centre of bha-cakra, ravi is at the 
centre of five tara graha. Hence at full circle (12 



I 



.•j 



True Planets 161 

rasi) or half circle difference, earth is in same line 
a s sun and the planet. 

Verse 19-22 : At 3 rasis after cakra or 
cakrardha (90° after 360° or 180° - i.e. 90° or 270°) 
i.e. at the end of odd quadrant, difference between 
planet's direction and sun's direction is maximum. 
Hence sighra paridhi is different at the end of odd 
and even quadrants (0°, 360"). 

Circle of naksatras (bhakaksa) is 360 times 
away from its centre earth, compared to distance 
of sun's orbit from earth. 

Division by this ratio (hara) 360 into degrees 
of 1 revolution (360°) we get 1° which is difference 
between sighra paridhi at the end of odd and even 
quadrants. 

I (author) wil explain the difference between 
mca and ucca paridhi. This can be seen direectly 
by observation, so presumption is not necessary. 

Explanations (1) Difference in sighra paridhi 
at end of odd and even quadrants is due to elliptical 
shape of planetary orbits. The difference depends 
on eccentricity of the orbit. It will be explained 
later on while explaining motion on sighra and 
manda paridhis. 

(2) The assumption is that difference in points 
of observation causes difference in paridhis. At 90° 
or 270° sighra kendra, difference is maximum due 
to sighra motion. Difference in heliocentric and 
geocentric position will be angle made by radius 
of sun orbit at distance of star circle. Difference of 
1° will be observed from 57.3 times the distance. 
This assumption will be correct if stars orbit is 
considered 60 times away compared to earth's orbit. 



162 Siddhdnta Darpana 

This figure has been accepted by Aryabhata, Surya 
siddhanta and all others. 360 times the distance 
will give less than 176 difference. Reasoning is 
wrong in both ways - about reason of difference 
in sighraparidhi and about the angle of difference. 

(3) Siddhanta darpana has taken sun's dis- 
tance about 11 times the figure accepted in classical 
siddhantas (based on 72,000 yojana diameter in 
Atharva veda is stead of 6,500 yojana in siddhanta). 
He has increased distance of stars further 6 times. 
Even after increase of about 66 times, it is still 
highly under estimated. Even the nearest star (4.4 
light years) is 2,80,000 times the distance of sun. 
Rohini at 14 lakh times and svati about 87 lakh 
times the distance are other nearest stars. 

Verses 23-27 * Attraction of ucca - Planets 
starting from ravi (all) are attracted by gods named 
their mandocca by chord of air (invisible force of 
attraction to an imaginary point mandocca). Hence 
true planets (spasta graha) are always deviated 
towards mandocca from their mean position (mad- 
hyama graha). 

Planets with slow motion of own and attracted 
by their mandocca, also move under influence of 
pravaha from east to west (due to daily motion of 
earth). 

(From Surya siddhanta) - Planets being always 
(day and night) under attraction of their ucca, move 
in different ways - sometimes east or west. When 
ucca is in east semidrcle of the planet, ucca pulls , 
the planet towards east. When ucca is in western -j 
semi circle, it pulls towards west. Direction of ucca 
on a circle is always same, but it is called east in 



True Planets 163 

one semi circle and west in the other. When the 
planets move towards east under attraction of ucca, 
the deviation is positive and in west it is negative. 
(Normal motion of planets relative to stars is 
towards east hence deviation in same east direction 
is added and in opposite direction, substracted.). 

Explanation - (1) Actual orbits of ravi and 
candra are elliptical round the earth (relative motion 
of ravi). In such an orbit earth is not at centre but 
on a focus. Thus centre of planetary motion is 
deviated towards farther end of major axis called 
mandocca. It is called so because at this position 
candra is farthest and hence slowest (ucca and 
manda). To a first approximation mean planet 
moves in a circle round earth at the focus. Next 
approximation assumes motion in an eccentric circle 
with centre at centre of the elliptic orbit.All the 
points of this circle are thus deviated from 
corresponding position of madhya graha towards 
mandocca. Hence mandocca appears to attract the 
planet. More accurately, mandocca doesn't attract 
because in such case, speed of the planet will be 
increasing in that direction. It should be maximum 
at mandocca. But it is only a deviation or 
displacement towards mandocca. This will be 
explained mathematically while computing the 
corrections. 

(2) If we rotate along a circle/ after reaching 
mandocca, the movement will be away from it upto 
180° difference. In remaining half it will be towards 
mandocca. Since attraction is always towards 
niandocca, 0° to 180° circle after it is considered 
negative or western deviation. 



164 Siddhanta Darpana 



■■< 



Verses 28-33 - Viksepa and pata - Orbit of j 
planets starting from moon (except ravi) are at an j 
angle with the kranti vrtta (i.e. apparent orbit of j 
ravi). It meets kranti vrtta at the points, where the \ 
circles bisect each other (being great circles of a '; 
sphere). Half of the orbit is north of kranti vrtta 
(upward direction of right hand screw rotating in j 
orbit direction). Other half is south. At pata, the 
planets are on kranti vrtta; Away from pata they 
are deflected towards north or south slight from 
Kranti vrtta. Hence pata is considered reason of 
north or south deflection called viksepa (or sara). 
Sara is distance of perpendicular from graha on 
Kranti vrtta, measured in kala or minutes of angle). 
Foot of perpendicular is manda sphuta graha. 

(From Surya siddhanta) - When planet is 
ahead of its northern pata by 0° to 180 °, it is 
deflected northwards. (Hence this pata is called 
northern pata or pata, in short). Then pata is 
behind the graha and called in west. Pata in east 
(or before the graha) deflects it towards south. 

Budha and sukra revolve with sun (being in 
inner orbit). Their sighrocca is the planet itself. So 
pata east from sighrocca causes south sara. In other 
half of orbit it is north sara. 

When pata is with graha or 180° away, graha 
is on one of the pata points and hence on the 
kranti vrtta. There is no sara in that position. When 
difference between planet and pata is 90° or 270 
(end of odd quadrants), sara is maximum (parama). 

Mean parama sara of planets is given below 
(compard with modern values given in introduc- 
tion) 



True Planets 






ibl 


planet 


Sara in Kala 


Degree 


Modern Value 


Candra 


309 


S'9 7 


5°8'42" 


Mangala 


111 


1*51' 


1*51'0" 


Budha 


164 


2°44' 


7*0*14" 


Brhaspati 


78 


1*18' 


1*18'21" 


Sukra 


148 


2*28' 


3*23'39" 


Sani 


149 


2 C 29 / 


2°29 / 25" 



Notes (1) Siddhanta darpana figures are an 
imprpvement over previous siddhanta books ac- 
cording to comparative chart given below. Except 
two inner planets, this compares well with modern 

values. 



(2) Pata according to other texts 



■ 


Surya 


Brahma 


sphuta 


Mafia 


Ptolemy 




Siddhanta Siddhanta & 


siddhanta 








Siddhanta Siromani 

• 








Candra 


4°3(T 


4°30' 




4*30' 




5*0' 


Mangala 


1"30' 


1*50' 




1*46' 




l'C 


Budha 


2*0' 


2*32' 




2° 18' 




7*0' 


Guru 


VV 


1*16' 




1*14' 




1*30' 


Sukra 


2*0' 


2*16' 




2*10' 




3*30' 


Sani 


2°cr 


2*10' 




2*10' 




2*30' 



(3) While figures of siddhanta darpana for 
moon and outer planets are very accurate, it looks 
wrong for budha and sukra compared to modern 
figures. Reason is that the modern figures are 
heliocentric whereas these are geocentric. 

In this figure 9, E is earth centre, S is sun 
centre and ESA is Kranti Vrtta, cut by plane 
perpendicular to budha orbit. This plane cuts 



166 Siddhanta Darpana 

B' 




B 
Figure 9 - Pata of inner planet 
budha orbit at points B and B'. As seen from sun, 
budha orbit makes angle BSE or B'SA with kranti 
vrtta. From sun this is parama sara. As seen from 
earth, parama sara is ZB'ES or Z.BES (almost equal 
because budha orbit is small). 

If ES is taken as 1, then mean radius of budha 
orbit SB = SB' = 0.3871. 

sin L B E S _ sin L B S E 

BS " BE 

B S 
or sin L B E S = =-= x sin L B S E 

d h 

0.3871 « 

= — - — x sin 7° o' 14" 

= .3871 x .1219= 0.0 472 
or L B E S = V 42' 

which is very close to value for budha in this 
book (2° 44') 

Relative distance of Sukra is 0.7233 hence its 
angle is given by (as seen from earth) 

Sin 6 = .7233 x sin 3°23'37" 

= .7233 x .0592 = .0428 

or = 2° 27' which is only V less than 
siddhanta darpana. 

Verses 34-41 - Types of planetary motion - 

Motion of planets as seen from earth's surface is 
called sphutagati. Sphuta gati is of three types - 



True Planets 167 

forward (prak or margi)) gati, reverse (vakri) gati 
an d zero motion (sunya gati). 

Forward motion is of five types, reverse 
motion of two types and one zero speed - these 
are eight types of motions of planets starting from 
mangala (Ravi and candra have no reverse motion). 
From Surya siddhanta - Eight gatis are named 
. 1. Vakra 2. Anuvakra, 3. Vikala, 4. manda, 5. 
mandatara 6. sama, 7. slghra and 8. Atisighra. 

Reverse motion starts reducing in later half, 
it is called vakra. At the start of reverse motion it 
is increasing. Then it is called anuvakra. 

When forward motion is less than mean speed 
and is still decreasing, it is called mandatara,. when 
it is increasing, it is called manda. 

Spasta gati equal to mean speed is called sama. 
Spasta gati more than mean speed and still 
increasing is called slghratara. When decreasing it 
is called slghra. 

Ravi and candra are affected only by mandoc- 
ca (not slghrocca). They have only five types of 
speeds - 1. manda, 2. mandatara, 3. sama, 4. slghra 
and 5. sighratara. Their meanings are explained 
above 

Verse 42 - Sphuta method - (From surya 
siddhanta) Now, I tell will respect, the method of 
making a graha sphuta by calculation, where mean 
planets arrive due to 8 types of speeds at the 
observed place, (drk-tulyata = calculation equal to 
observation). 

Verses 43-46 - Explanation of arc and its sine 
- Now, I tell the method of calculating skue and 



168 Siddhdnta Darpana 

arc, which is used in many sciences and by 
knowing which, people get the title of acarya 
(doctorate). 

As a cloth is interspersed with threads, a gola 
(sphere or its circles) is also mixed up with sines 
and versed sines, (sine can be found between any 
two points of a sphere or circle and hence they 
are infinite in number). 

To find the sine (jya) of a radius inclined with 
starting radius at 0°, we make a jya of same arc 
in opposite direction. Graha is on top of jyarddha 
(end of radius vector) with which calculation is 
made. It is also called jya in short. Half part of a 
circle or full revolution (bhacakra) looks like a bow 
(capa). The bisecting line of circle passes through 
its centre and is called diameter (vyasa). 

Verses 47-54 - Method of Calculating sines - 

For 3 raSi (90°)jya, Kotijya have extreme values. 
Jya of 3 rasi passes through centre and is equal to 
radius. Then it has greatest value, Kotijya is 
distance of this jya from centre and is here (for 
90°). 

96th part of a circumference is very small and 
almost a straight line. Hence it is almost equal to 
its jya. Thus 1/8 part of a rasi lipta (1800) i.e. 1/96 
of circle is 225 and is equal to first jya. 

First jya is first khandantara (difference) (i.e. 

Jya of 225' - jya of ; )i To find the second 

Khandantara (2nd jya - 1st jya), 1st jya is divided 

by itself and result (1) is deducted from 1st jya 

225 
(225 - 225 = 224). Result is 2nd khandantara. 

Add 2nd khandantara in 1st jya to get the 
2nd jya (i.e. sine of 2x225' arc). 2nd jya is again 



True Planets 1*9 

divided by 1st khandantara. If remainder is more 
than half of 225 then it is ommitted. Quotient 
deducted from 2nd khandantara gives 3rd 
Idiandantara. 

3rd Khandantara addeed to 2nd jya, we get 
the 3rd jya. Similarly jya pinda (quantity) is divided 
by 1st jya and substracted from its khandantara to 
give next khandantara. This way we get the jya 
of 1st to 24th jya pinda in lipta. In dividing 
6,7,12,15,17,20,21 jya pinda, reemainder is more 
than half of divider 225. But we still omit it without 
adding 1 to quotient because Brahma had told so 
to Narada. 

Notes : (1) Narada Purana also gives a 
complete summary of astronomy and astrology. In 
the chapter describing calculation of sines, no such 
explanation from Brahma has been given as stated 
here. However, the stated values have been given 
which means the same thing. There is no dialogue 
from Brahma in the chapter, but he is considered 
original source of the knowledge. 

(2) Increase in sines is proportional to its 

d 
differential coefficient thus — (sin x) = cos x is 

ax 

proportional to 1st difference (khandantara) Change 

in difference itself; i.e. 2nd diff is proportional to 

2nd differential coefficient. Thus 2nd difference = 

d 2 d (cos x) . . . 

"Tj (sinx) = — H = - sin x. (proportional) 

Let 1 part be P = 225'. Hence sin x = Sin nP, 
n=no. of parts. 

Here jya = R sin x where R is radius equal 
to 3438 kala. 

w 

First difference = Ai (or 1st khandantafa). 



170 Siddhanta Darpana 

d 
Ai= .— (R sin x) . Sin P = R (Cos x) Sin P 

= R sinP. 

c j jvcf ■ r* & R<5x.sinx 

Second difference S = -7- (R cos x) = - 

dx x ' R 

Negative sign means that the first differences 
(khandantaras) are decreasing with increasing angle 
and 2nd differences are proportional to jya (R sin 
x). It is to be divided by R to get modern sine. 

(R cos x . d x) _ -R sins x . dx 2 

R - tf 

a* onO -. - , d ** 225X 225 

At x = 90 it is equal — — = — rrrr 

M R 3438 

= 14 / 43"30 / ". This has been explained by Sri 
Ranganatha in his tika on Surya siddhanta called 
Gudhartha - Prakasika. But he has taken this as 
3438/225 = \5'\6"^'" by mistake. Even this is 
approximate and correct value is 14'47" 

(3) Sri Bapudeva Sastri has given the following 
proof of the formula in his English translation of 
Surya siddhanta 

Ai = Sin P - Sin 0° 

A 2 == Sin 2 P - Sin P 
" A 3 = Sin 3 P - Sin 2 P 

An = - Sin n P - Sin (n-1) P 

An+1 = Sin (n+1) P - Sin n P 
Then Ai - A 2 = 2 sin P - sin 2 P 
= = 2 sinP - 2 sin P cos P 
= 2Sin P (1 - cos P) = 2 Sin P. ver sin P 
(versed sine = lrcosine = utkrama jya.) 

A 2 ~ A 3 = 2 Sin 2 P - Sin P - Sin 3 P 



True Planets I 71 

= 2 sin 2 P - sin P - (3 Sin P - 4 Sin 3 P) 
= 2 sin 2 P - 4 Sin P + 4 Sin 3 P 
= 2 sin 2 P - 4 Sin P (1 - Sin 2 P) 
= 2 Sin 2 P-4 Sin P. Cos 2 P 
= 2 Sin 2 P - (2 Sin P Cos P) 2 cos P 
= 2 Sin 2 P (1 - cos P) 
= 2 sin 2 P. versin P 
A 3 - A4 = 2 sin 3 P - Sin 2 P - Sin 4 P 

= 2 Sin 3 P - 2 Sin 3 P. Cos P 

= 2 Sin 3 P (1 - Cos P) = 2 Sin 3 P versinP 

An- A n + i = 2 Sin n P-Sin (n-1) P - Sin (n+l)P 

= 2 Sin n P - 2 Sin n P Cos P 

= 2 Sin n P (1 - Cos P) = 2 Sin n P versinP. 

Adding the above equations, we get 

Ai - A n + 1 = 2 versin P (Sin P + Sin 2 P + 
Sin 3 P + — Sin n P) 

But Ai - A n + 1 = Sin P + Sin n P - Sin (n+1) P 

Hence, Sin P + Sin,<nP - Sin (n+1) P = 

= 2 ver sin P (SinP + Sin2P + + Sin 

nP) 

or Sin (n+1) P = Sin nP + Sin P-2 versin P 
(Sin P + Sin 2 P + — Sin nP)) 

Here P = 3 a 45' = 225' 
*. 2 versin P = 2 ver sin 225' = 2 (1-Cos 225') 

2(1 -0.9978) = 2X0.0022 = ^ = ^ = ^ 

approx. 

Thus sin (n+1) P = Sin nP + Sin P - 1/225 x 
(Sin P + Sin2P+ -- Sin nP) 



172 



Siddhdnta Darpana 



This is the formula for finding (n+l)th sin 
from nth sin i..e sin np, First khandatara is added 
and sum of pervious sines divided by (225) is 
substracted. 

(4) Bhaskara II has explained that the sines 
were found by constructing regular polygons of 
increasing number of sides in a circle. 

Aryabhata has indicated the geometrical 
method for finding sines for 12 divisions of a right 
angle (7°30 / each) in a circle of radius R = 3438'. 
(Method is explained by Prof. Kripa Sankara Sukla). 

Let figure 10 represent a circle of radius R = 
3438'. Divide the quadrant into two at T (45°) each. 
Trisect TA into TB, BR, RA (15° each), RA into 

L 




Figure 10 - Geometrical method of sine table 

two (RQ, QA, 7-1/2° each). Mark off AL = 30°. 
Join LB. This is equal to R and denotes chord 60". 



2^ 
R sin 30" = — = m^ 



1° 



This is the 4th sine, in the 7 ■=- table to" be 

2 

computed New, from right angled A OMB, 



True Planets 173 

OM = V R 2 _ (H / 2 )2 = v1>R = 2978 

This is R sin 60°, i.e. the eighth R sine 
Now from rt angle AAMB 

AB = V (R sin 3q0 f + (R vers 3q0 f 

= V(i7i9)2 + (460) 2 = 1780 
This is chord 30°. Half of this i.e. AN, is R 
sin 15° 

Thus R sin 15° = 890' 
This is second R sine 
Now from rt angle AANO 



ON = V A0 2 - AN 2 ) = V R 2 _ (R sin ^ = 3321 

This is R sin 75° = the tenth R sine 

Now in rt AANR,, where R is mid-point of 
arc AB, we have 

AR 



= V AN 2 + NR 2 = ^(R sin 15' ) 2 + (r versl5° ) 2 

^890 2 + 117 2 - = 898' 

This is chord 15°. Half of this i.e. AS is 
R sin 7°30' 

Thus, R sin 7°30' = 449 / 

This is the first R sine 

Now, in rt A ASO, OS 

= V R 2_ (Rsin ,70 30) 2 = 3409' 

This is 11th R sine for angle 82°30 / 
Now, R ver s 75° = R-R sin 15°, s o that 

chord 75" = V( R sin7 5 ) 2 + (Rvers75° ) 2 = 4186 ' 

Half of this 2093' is R sin 37° 30' (fifth R sine). 



174 Siddhanta Darpana 

Now, R sin 52-3C = VP~T^~~"^1^=2728 (7th 
R sine) 

In semi square AOD, OA = OD=R 

so AD = V2 R = 4862' 

This is chord 90°. Half of this 2431° (i.e. AP) 
= R sin 45° 

In A AFT, AT = V (R sin 45 <> f + (r vers ^2 , 2630- 

This is chord 45°. Half of this is R sin 22*30' 
= 1315' 

(3rd R sine)) 

R sin 67-30' = V R 2 _ (R sin 22 ° 30) 2 = 3177' 

(ninth R sine). 

These are 12 R sines. By finding chord of 
7° 30' arc we can find R sines of 3° 45' intervals 
also. 

(5) More accurate method is to calculate sine 
by infinite convergent series. 

Sin 0= 6 - 2- + ^- + 

Z3 Z.5 

where 6 is expresseed in radians (arc/radius) 
and is betweeen 0° and 90°. 

verses 55-66 — All the verses are quoted from 
Surya siddhanta. 

Verses 55 - 60 - These tell the values of 24 
R sines at intervals of 3° 45' in kalas. Next verses 
give values of utkrama jya = R (1-cos 0) 



No. 


Arc 


R 


Modern 


Diffe 


Vers Diff. 


Modern valul 






sines 


values 


rences 




radius=l 1 

1 


1 


225' 


225' 


224.856 


225 


7 


7 


.0022 ] 


2 


450' 


449' 


448.749 


224 


29 


22 


.0086 \ 


3 


675' 


671' 


670.720 


222 


66 


37 


.0192 


4 


900' 


890' 


889.820 


219 


117 


51 


.0341 I 


5 


1125' 


1105' 


1105.109 


215 


182 


65 





True Planets 175 



6 


1350' 


1315* 


1315.666 


210 


261 


79 


.0761 


7 


1575' 


1520' 


1520.589 


205 


354 


93 


.1031 


8 


1800' 


1719' 


1719.000 


199 


460 


106 


.1340 


9 


2025' 


1910' 


1910.050 


191 


579 


119 


.1685 


10 


2250' 


2093' 


2092.922 


183 


710 


131 


.2066 


11 


2475' 


2267' 


2266.831 


174 


853 


143 


.2481 


12 


2700' 


2431' 


2431.033 


164 


1007 


154 


.2929 


13 


2925' 


2585' x 2584.825 


154 


1171 


164 


.3406 


14 


3150' 


2728' 


2727.549 


143 


1345 


174 


.3912 


15 


3375* 


2859' 


2858.592 


131 


1528 


183 


.4445 


16 


3600' 


2978' 


2977.395 


119 


1719 


191 


.5000 


17 * 


3825' 


3084' 


3083.448 


106 


1918 


199 


.5577 


18 


4050' 


3177' 


3176.298 


93 


2123 


205 


.6173 


19 


4275' 


3256' 


3255.546 


79 


2333 


210 


.6786 


20 


4500' 


3321' 


3320.853 


65 


2548 


215 


.7412 


21 


4725' 


3372' 


3371.940 


51 


2767 


219 


.8049 


22 


4950' 


3409' 


3408.588 


37 


2989 


222 


.8695 


23 


5175' 


3431' 


3430.639 


22 


3213 


224 


.9346 


24 


5400' 


3438 


3438.000 


7 


3438 


225 


1.0000 



Notes (1) Difference for versed sines are in 
opposite order and they need not be calculated. 
From them versed sines are calculated. 

(2) Modern values of sin, cos and other ratios 
are calculated for radius 1. Hence, for calculating 
Indian sines they are to be multiplied by radius. 

(3) Madhava method for calculation upto 9 
decimal places - This has been quoted by 
Nflakantha in his commentary on Aryabhatfya. His 
sentences indicating calculation parameters have 
been quoted by Sankara in his commentary on 
Tantra sangraha by NHakantha. Original book of 



n 



p 

.*:.< 



Siddhdnta Darpana 

Madhava is not available. He must have useJ 
infinite series and then formed the simplified ruleJ 
expressed by verses in 'Katapayadi' form. ^? 

Method for sines - Place the expression*! 
0'0"44"', 0'33"6"', 16'5"41"', 273'57"47"', anl 
2220'39"40"' - five numbers from below upwards^ 
Multiply the lowest by the square of the choseij 
arc and divide by R 2 (i.e. 2,91,60,000 = 5400 2 )! 
Substract the quotient from expression just above}, 
Continue this operation through all the expression! 
above. The remainder got at last operation is m 
be multiplied by the cube of the chosen arc an 
divided by R 3 (i.e., 157,46,40,00,000). Substract t 
quotient from the chosen arc to get its R sine. 

Method for versed sines - Place the si 
expressions - , /, 6' // ,0 / 5'12 /, / 3'9"37"' 71'43"24 ,i 
872'3"5'" and 4241'9"0"' from below' upward 
Multiply the lowest by the square of the chose 
arc and divide by R 2 . Continue the operatio; 
through all the opeations above. The last quotie* 
will be the versed sine of the chosen arc. Thi 
formula is based on series for sin upto term n 








Results 






No. 


Arc. 


R sine sine in decimal 


1 


225 


224'50"22"' 


.06540 


2 


450 


448 , 42"58 , " 


.13053 


3 


675 


670 , 40"ll m 


.19509 


4 


900 


889'45"15"' 


.25882 


5 


1125 


1105T'39"' 


.32144 


6 


1350 


1315 , 34 ,, 7"' 


.38268 


7 


1575 


1520'28 , '35" 


.44229 



Modei 
Value 

.06540 

.13053 

.19509 

.25882 

.32144 

.38268 

.44228 



True Planets 



177 



8 


1800 


9 


2025 


10 


2250 


11 


2475 


12 


2700 


13 


2925 


14 


3150 


15 


3375 


16 


3600 


17 


3835 


18 


4050 


19 


4215 


20 


4500 


21 


4725 


22 


4950' 


23 


5175' 


23 


5400* 



.50000 .50000 
.55557 .55558 



1718 , 52"24 , " 
1909 , 54"35"' 
"2092'46 w 3 m 
2266'39"50 m 
2430'51 ,, 15 m 
2584 , 38 M 6 m 
2727*20"52 m 
2858'22"55 m 
2977'10"34 m 
3083'13 w 17 m 
3176'3"50 ,n 
3235 , 18"22" 
3320'36"30" ? 
3371 , 41"29 m 
3408 , 20 w ll m 
3430'23"ll m 
3437'44»48 m 

Verses giving value of constants in katapayadi 
constants for sine - (with method) 

fa*c*llPcW: WF$ t #w|ijM^q4«T *ftfc s^^d: II 

Constants and method for versine - 
'#T:' ^ftfr?p:' ^^^Tf^gT ^T^T^^; 'TOf ^w4t ' 

3TT«R^T^TjfnRfT <?% «Fpi: ^H fa^lfcW: 

Sine table in paras - 



,60876 .60876 

.65934 .65934 

.70711 .70711 

.75184 .75184 

.79335 .79335 

.83147 .83146 

.86603 .86603 

.89687 .89688 

.92388 .92388 

.94693 .94692 

.96593 .96593 

.98079 .98079 

.99144 .99144 

.99785 .99785 
1.000001.00000 



178 Siddhdnta Darpat 

'^^^^''qm^^«^' im 11 

'^^^■^^^"Hftwereft^'.ll^ll 
cicH<lR^^pm^^TT^r5^Tn«T%f^: |V9 I 

Proof of the method for sines - 

Madhava has used the infinite convergent 
series for sine for 6 expressed in radians (betweei 
and nil) 

L3 L5 [7 L9 |n 
(Terms upto 6 n have been used for desired 
accuracy) 

e 3 e 5 e 7 e 9 



h 



1 



or sin# = $ + 



6 120 5040 3, 62, 880 

B n 1 



3 99, 16, 800 " (1 ) 

Constants of Madhava expressed in para are 



'H 



frue Pl anets 179 

(1) 79,94,380 = Ai (2) 9,86,267 = A 2 (3) 57,941 
= A3 (4) 1,986 = A4 (5) 44 = A 5 from up to down 
order. 

Let x is the arc length in minutes (kala). 

Then 6 converted to degree (x 180/n; ) and 

in minutes becomes x 

180 10800 

orx = 6x — x60 = -^-e 

At each stage we take its square and divide 
by R 2 . (R = 5400) i.e. multiply by a 2 where a = 
x_ 
R 

x 10800 n 1 20 

_ _ — — x 6 x — — - = — « a (2) 

R n 5400 » w - 

After multiplying 5th quantity by a and 

substracting from 4th, we get A4 - a A5 

2 
Multiply this by a and substract from A3, we 

get A3 - a A4 + a A5 

2 
Multiply by a and substract from A2, we get 

A2 - a 2 A3 + a A4 - a 6 A5 

Multiply this by a and substract from Ai, we 
get Ai - a 5 A2 + a* A3 - a 6 A 4 + a 8 A5 

Multiply by a 3 and substract from arc x - 

10800 

— jf- we get 

x - a 3 Ai + a 5 A 2 - a 7 A 3 + a 9 A4 - a 11 A 5 
This is value of sin 6 in arc length of minutes. 

To get it in ratio for radius 1 we have to divide 

* by 10800/;*. Then 

Sin 6 = j^- ( X . a 3 Ai + a 5 A 2 - a 7 A3 + a 9 A* - a n A 5 ) 

- - (3) 



180 Siddhanta Darpana 

First term ■■ tq§qq x x = 6 radians 

«. ■ . % fa) 3 79,94,380 

Second term = ID g gg x \^J ^ 

(A is divided by 3600 to convert it in minutes) 

- — approx. (taking n = 3.151926 ) 

Similarly we get all the terms of series (1) 
from formula (3) by calculations. 

(4) Vatesvara has used 96 divisions of a 1 
quadrant, each division being 56'15" of arc. He hasj 
given values of R sine and R versine in seconds! 
of arc. 

i 

Murusvara has taken radius length as 191 and j 
at 1" intervals, given the values of R sines uptoj 
4th division of a degree (upto 1/60x60) of a second). | 
Kamalakara and Jagannatha Samrata both have! 
taken radius of length 60 and given values upto j 
5th division of a degree. They have taken intervals 
of 1" and 1/2" respectively. 

(5) Direct computation of R sines - Madhava 
formula an be used fbr any angle for calculation 
upto 9 decimal places. Bhaskara I, Brahmagupta, 
Vatesvara and Srfpati have given formulas for direct 
calculation. All of their formula are equivalent to; 

the following expression. 

e (180 - 0) 
Sin = — — — — ; $ in degrees. 

10125.- - 0(180-0) 

While explaining calculation of sine ratios 
- upapatti vasana), Bhaskara n> has given two foi 
of formula which Reduce to the same expression. 

Proof : In Figure 11, C A is diameter of a 






True Planets 



circle of radius R 

Arc AB = 0° 

and BD = R sin 

1 

Area ABC = - AB. 



BC 



2S1 



Also, Area ABC = - AC. BD 



So 



AC 



BD AB . BC 



so that 



AC 



Let 



BD (Arc AB) (arc BC) 
1 xAC 

BD ~" f*rr Am r*^nr\ + Y 



2xR 



(arc AB) (arcBC) 



(180 - 0) 



zr^ + y 




or 



Rsin0 



Figure 11 

2xR + g(180~-fl).-y 
0~(m - 0) 



or P-^p- ^(180-6) 

2xR + 19(180 -0)y 



(1) 



182 Siddhdnta Darpana 

■ « a 1« 30 x 150 

Putting = 30°,-R = 



2xR + 30 x 150 y 

9000 
or 2R + 4500 y = -z- (2) 

Putting 6 = 90°, \\ 

8100 
2xR+ 8100 y = — - - - ■- - (3) 

1 ■ „ 4050& 

Form (2) and (3) y = - — and 2xR = — -=- 

v ' v ' J 4R 4 R h 

■ 0(180-©) ' I 

Hence from (1) Sin © =. — — "i 

10125-7 6 (180 -6) I 

Alternate proof : 

a + bA .+ cA 2 I 

Let sin A = — r ti 

A + BA + CA 2 I 

where A is in radians and corresponds to flf 
degree ','*'* 

Putting A = 0, a = 

"■■■" '„ '& 

Putting A = Jt, b + jt c = So c = -^ 

bA (n '— A)/tt 
Thus sin A = A + BA +CA z 

Since Sin A = sin ( jt - A ), 

bA (x - A)/jt _ frA (x - X)/x 

A + BA + cA 2 A + B(^ -A) + C (^ -A) 2 

or, A + BA + cA 2 = A + B (x - A) + c (jt - A) 2 
or, B (2A - x) = C x (x - 2A) 

or, C = - - 



■%i 



■s 



True Planets 183 

Therefore, Sin A = - — .**, 1N 

A 7i + B A (it - A) 

Putting A ■» - jr (Sin A '*-). 



A« + B 7 «(jr-f) a5 '2b.f (»-5) 



5;r 2 B 10;r 2 b 

cr,A» + — -■ = -3P (4) 

Putting A - I (Sin A = 1) 

A*: + B 2 (* - 2) = — (jt - j) 

. 1 2 „ 1 2 

or A jz + -7T B = - jT b (5) 
4 4 

From (4) and (5), B = - 7 b, A = -^ 

4 16 

Therefore, Sin A = —7 — — — ) 

5 n* -AX (x-X) f 

71 

whereA = — 

Verses 67-70 - Jya of bhuja and koti - Madhya 
graha substractecl from mandocca gives manda 
kendra and from sighrocca, it gives slghra kendra. 
From these quantities, bhuja and koti jya are 
calculated. 

In odd quadrant (visama pada), jya of passed 
arc is bhuja jya and remaining arc gives koti jya. 
Irt even (sama) quadrant, remaining arc gives bhuja 
jya, and passed arc gives koti jya. (Quotation of 
Surya siddhanta ends with verse 68). 

This means that if the kendra (manda or 
sighra) is less than 3 rasi (90°), then its jya is bhuja 
J 7 a * If li is between 3 to 6 rasi, then it is substraeted 
from 6 rasi. Jya of the balance arc is bhuja jya. 



184 Siddhanta Darpana 

When kendra is between 6 to 9 rasi, we deduct 6 
rasi from kendra. Jya of the balance arc is bhuja 
jya. If kendra is between 9 to 12 rasi, it is deducted 
from 12 rasi. Jya of balance arc is bhuja jya. If the 
quantity, from which deduction is to be done is 
smaller, then 1 rotation of 12 rasi (360°) is added 
to it. 

Notes (1) The rulers very simple and needs 
no explanation, if figure 12 is seen. ANBU is kranti 
vrtta in which U is ucca (manda or sighra). The 
planet moves in anti clock wise direction shown 
by arrow. Mi, M2, M 3 and M4 are positions of the 
planet in 1st, 2nd, 3rd and 4th quadrants from 
ucca position. Displacement of the planet along AB 
line is indicated by perpendiculars from M on AB 
i.e. parallel to NU line. 

Thus in quadrant 1, Mi,U is the passed arc 
of angle Mi,DU. Its sine is Mi, Yi, or OXi, called 
bhujajya. Similarly bhujajya in 2nd, 3rd, and 4th 
quadrants are OX 2 , OX 3/ OX4. . 

Kotijya is jya of complementary angle (90° - 
angle) or cosine in modern terms. It is indicated 



8 




Figure 12 - B4hu and Kotijya 



True Planets 185 

by displacements along UN line. Kotijya in the 
quadrants are OYi, OY 2 , QY 3/ OY 4 . 

(2) Manda Kendra = Mandocca - Madhya 
graha. 

Sighra kendra = Sighrocca - Madhya graha 

Both are definitions and need no further 
comment. • 

Verses - 71-72 - Method of finding jya and 
utkrama jya of any angle - (Quotation from Surya 
siddhanta) 

From the chart we get sines and versines, 
only for angles which are multiples of 3° 45' (225'). 
Finding values for any intermediate angle is called 
interpolation.. Its method according to Surya 
siddhanta is 

— - — • . = Quotient (= completed parts 

of 225') 

+ remainder (angle lapsed in next part) 

Jya of angle = Jya of previous part + 
remainder/225 x (Jya of current part - Jya of 
previous part.) 

Notes (1) This is simplest formula based on 
ratio and propotion i.e. rule of 3 (to find 4th 
unknown quantity). 

Difference in Jya of fractional part 

Angle of fractional part 

Difference in jya of completed part 
Angle of 1 part (225') * 



*£6 Siddhanta Darpana 

This assumes proportional variation of sine? 
difference. This is called linear variation or linear 
interpolation. 

(2) For small divisions of 3° 45' each; linear 
formula is sufficient to get accuracy upto a minute. 

If divisions are of 10° each, then sine 
difference doesn't increase proportionate to increase 
in angle. For such interpolation, Bhaskara II has 
given quadratic formula - 

y = yo +- (yi - yo) + ,' (72 - 2yi + yo ) 

where h = intervals (10') at which sines have 
been calculated; x is increase in angle; yo, yi, and 
V2 are difference of sine for successive parts. 

This formula was first stated by Brahmagupta. 
Vatesvara has given several forms of the formula. 
For R sines upto seconds, quadratic formula is 
needed. Brahmagupta expression for 225' intervals 
is - 

R sin (225't+0') - - - (where 0' < 225', t is an 
integer) 

- sum of t R sine - differences 

ej_ r t th R sin diff + (t + l)th R sin diff 
225 L .2 
6 ' t th R sin diff + (t + l)th R sin diff 
-^5~ ^2~ ~ 1(1) 

= sum of R sine differences 

+ ^ (t + 1)* R sin difference 



+ -r 



1 0' {&' 



2 225 



diff.] 



225 



— 1 



[ ( t+1) th R sin diff. - t th R sin 
(2) 



(2) is equivalent to quadratic formula. 



187 
True Planets 

(3) Madhava's formula - If t is a positive 
integer and < 225', then 

R sin (225' t + 0) = sum of t R sin - diff 

6 x [R cos 225 (t + 1) + R cos (225 t)] 



+ 



2R 



This has been quoted by NHakantha in his 
commentary on Aryabhatiya. 

Verses 73-74 : Finding arc from any given 
jya - We substract the greatest jya pinda/ smaUer 
than given jya, from the given jya. The difference 
of jya is divided by difference of the jya pinda 
substracted and next bigger jya pinda. It is 
multiplied, by 225' and result is added to the 
completed arc. 

Notes - (1) This is again linear interpolation 
whose proof is similar to reverse process of finding 
sine of a required arc. 

(2) There is a similar quadiratic formula for 
this process also. 

(3) Geometrical proof of the formula (linear 
and quadratic) — 

Figure 13 shows the graha of R sine with its 
arc (or angle) On OX axis xo, xi x 2 are points 
indicating angles or arcs where value of sine is 




.XX, x, 

Figure 13 - Finding sine or arc for intermediate values 



188 Siddhanta Darpana 

known. R sine at these points is no, n t , n 2 . We 
have to find value of R sine x = D on the graph 
when Ox is known in interval (1,2) Alternatively, 
if D is known, we have to find x point. 

In linear interpolation, we assume that value 
of R sine changes along the straight line AC from 
no to ni. For small intervals it is almost same as 
curve AC; Dx is perpendicular on X axis cutting 
AC at D' and AB at E. Thus D' is a good 
approximation (linear) for D on the curve. 

Now D'E = D'x - Ax = increase in value of 
R sine for desired angle. 

C B = increase in Rsine over interval (0, 1) 

" D'E - AE 
HenCe lc-= AH"'.., 

When x is known, we know AE, AB = 225' 
and BC is already known (value of R sines). Then 
fourth quantity D'E can be known. Required sine 
is D'x = D'E + A x which gives the formula. 

D ' E x AB 
Similarly Xqx = AE = — - can be known 

and O x = Ox + xxo 

* . 

Quadratic interpolation - ni-rto is difference 
between the interval. (n 2 -ni) is difference in next 
interval. 

Average rate of change is 

(ni - n 0) + (n 2 - ni) 

2 
Increase in rate of change within the interval 

AE (m - no) - (w2 - «i) 
" AB 2 



True Planets 189 

Second term is the extra term for quadratic 
form of Brahmagupta. 

Verse 75 : Relation between sine and cosine 
, From square of radius, deduct the square of jya, 
take square root of the quantity* Result will be jya 
of koti of the angle (3 rasi - angle) or kotijya. 
Similarly, jya can be calculated from kotijya. 

Note - Cos (9O°-0) = Sin 6 or sin (9O°-0) = 
CosO 

Sin 2 6 >* Cos z fl = 1 

Hence V r2 _ R 2 sin 2 e s R Cos 6 

Verses 76 to 88 : True motion of mandocca 
by difference from parocca - 

Mandocca of mahgala, budha and sani and 
sighrocca of budha move both forward and 
backwards as observed by me (author). To find 
their true motion another entity (devata) called 
'Parocca' has been assumed which affects these 
points (verse 76). 

Parocca of mahgala (its mandocca) is at 6 rasi 
(180°) difference from madhyama surya. Parocca of 
budha mandocca is its Sighrocca. Likewise, man- 
docca of budha is parocca of budha's sighrocca. 

Parocca of sard's mandocca lies at 18° less 
from madhyama sani. Parocca also is an invisible 
form of Kala like mandocca. 

Similarly there are many small planets in the 
sky. Around planets, satelites move in circular 
orbits. Motion of these small planets and satellites 
can be seen only with instruments (telescope). They 
also move from west to east. They have not been 
described in Brahma and surya siddhanta, hence 
they are not been explained here. * 



igO Siddhanta Darpana 

Substract parocca from iighrocca and mandoc- 
ca of budha and from mandocca of sani. R sine 
of the resulting angle is multiplied by 680 for budha 
and by 300 for sani. Product is divided by radius 
(3438). Result will be in lipta. If Para kendra 
(distance of manda or sighra from Parocca) is 
between 0° to 180° /it is added in mandocca and 
for 180° to 360% it is substracted. For slghrocca of 
budha, its opposite procedure is followed (substrac- 
tion for parakendra in 0° to 180° and addition 
other wise. 

If sighra kendra of mangala is in sue rasi's 
starting from makara (i.e. from 270° to 90 V then 
sphuta sanskara is not needed for its mandocca. 
If sighra kendra is in six rasis beginning with karka 
(i.e. 90° to 270°) then mean mandocca will be 
corrected to find the true value. Mandocca of 
mangala for previous year (almost same for current 
year) is substracted from its parocca (mean smv + 
180°). Result is its para-kendra. R sine of para 
kendra is multiplied by 450 and divided by 3438 
= radius. Result is multiplied by kotijya of first 
sighra kendra in kala and then divided by kala of 
3 rasi (5400). As before, result is added in 1st six 
rasis of para kendra and substracted in other rasis 
from mandocca of mangala. 

For daily motion of mandocca - Substract 
mandocca from parocca of mangala (i.e. 180° + 
mean sun). Its kotijya is multiplied by 450 and 
divided by radius (3438). This will be kotiphala for 
correction of mandocca speed. If parakendra is in 
0" to 180°, kotiphala is added to sighra kotijya and 
substracted for parakendra between 180° to 360°. 
Result is multiplied by mean daily motion of sun 



True Planets 191 

# ■ 

and divided by radius (3438). Result will be daily 
motion of mangala mandocca due to its parocca. 

Similarly for sani, find kotijya of its difference 
f parocca and mandocca. Multiply it by 300 and 
divide by radius 3438. Result is kotiphala for 
correction. Kotiphala is multiplied by mean daily 
motion of sani and divided by radius 3438. This 
will be daily motion of sani mandocca as corrected 
for its parocca effect. 

For budha, Kotijya of its para kendra is 
multiplied by 680 and divided by radius 3438. 
Result will be Kotiphala for correction. Multiply 
this Kotiphala by daily mean motion of budha 
sighrocca and divide by radius 3438. This will be 
daily motion of true budha sighrocca. Correction 
in mandocca speed is addition for parocca kendra 
in 1st and 4th quodrant (270° -90°) and otherwise 
deducted. Correction in sighrocca is in opposite 
manner. 

Notes - (1) Author has not given reasons for 
such correction. His mention of observation of small 
planet by telescope indicates that these correction 
are based on some modern charts like Le - verrier's 
chart of 1850 or some nautical almanc available in 
his time. He has clearly mentioned that these have 
not been discussed in other siddhantas which 
indicates his corrections are adopted from some 
almanc or results of telescopic observation. 

(2) It is difficult to guess as to what correction 
was sought to be achieved by these methods. 
However, mathematical form of these formula will 
indicate the reasons of these corrections. 



*92 Siddhdnta Darpana 

(a) Definition - Parocca' is a mathematical | 
point in ecliptic from which deviation in mandocca I 
of mangala, budha and sani and sighrocca of budha \ 
can be calculated. 

(b) Parakendra (P) = Sighrocca or mandocca j 

- parocca. 

Parocca of Budha sighra = mandoca of budha 

Parocca of budha mandocca = sighrocca of 
budha 

Parocca of mangala mandocca = madhyama 
ravi + 180° 

Parocca of sani mandocca = madhyama sani 

- 18° 

(c) Madhyama mandocca of all the planets is 
given in madhyamadhikara. Sighra of inner planets 
budha and sukra are the planets themselves. Sighra j 
of outer planets mangala, guru, sani is mean sun. 4 
Thus the tara graha, affected by own orbit as well 
as earth orbit (or relative motion of sun) have 1 
sighrocca as the planet of smaller orbit (and hence 
of faster rotation). 

(d) Para kendra of Budha sighrocca (Bi) i.e. 
PBi, = Bi - B 2 where B 2 is mandocca of Budha. 

Para Kendra of B 2 (mandocca of Budha) 

PB 2 = B 2 - Bi 

Correction in mean Bi = 680' sin (Bi-B 2 ) 

Correction in mean B 2 = 680' sin (B 2 -Bi) 

Sin (Bi-B 2 ) is positive when PBi = B r B 2 is j 
between 0* and 180° then it is negative correction J 
For mandocca it is opposite. Hence for both budha | 
mandocca and sighrocca, correction is 



■i 



True Pla^ s 193 

680' Sin (B 2 -Ba) - - - (1) 

Parakendra of sani mandocca, Ps is 

S 2 - (madhyam Sani - 18*), S2 = mandocca of 

Sani 

or Ps = S2 + 18* - madhya sani 

= S 2 + 18 - S say - - - 

Correction to mandocca is 300' Sin Ps - - (2) 

Since it is to be added when Ps is between 
0° to 180° i.e. Sin Ps is + ve, the formula indicates 
correct sign. 

Sighra kendra of mangala = mean sun - mean 
mangala. Skm = S m - M 

When S Km is between 270° to 90% no 
correction is required. Correction to mandocca is 
done only when S Km is between 90° to 270° i.e. 
earth is on same side of sun as mangala. 

Amount of correction - 

Parakendra of mangala Pm = (Sm + 180").- 

M2 

where M2 is mandocca of mangala 

(This is opposite substraction of the earlier 
process) Correction in mangala mandocca 

450 sin Pm x cos S Km 
5400 

Sin Pm is + ve for Pm between 0° to 180° 
a nd it is added to mean mandocca. 

Speed of mandocca is obtained by obtaining 
the differential coefficient of the corrections. 

Position of true mandocca of mangala # 



194 



Siddhanta Darpan 



450 sin Pm x cos S Km 



5400 



450 



cos Pm x cos S Km d __ 

* -r~ rm 

5400 dt 

= speed of mean sun. 



Speed s 

dPm _ dSm 

IT ~ dt 

Thus speed of mangala mandocca 



>peed 



450 cos Pm . cos S Km 
5400 



-(fl 



Speed of Budha Slghra or Mandocca 

= 680' Cos (B 2 -Bi) - - - (5) 
Speed of Sard mandocca = 300' cos Ps (6) 
(3) Reason and assumptions of these corr 
tions - 

(a) Sard mandocca motion - Motion of 
mandocca cannot be observed in a life time or ev 
in a thousand years because it rotates only 39 tir 
in a kalpa. Hence it is not oscillatory motion 
mandocca which could be observed by the autho* 
This appears to be correction due to effect of guru' 
attraction on Sani motion. At the time epoch 
his observation after 1869 AD, guru was behin 
Stni for 5-6 years. Hence parocca of sani has bee 
assumed to be slightly less (18°) than mean sani 

(b) Mangala is corrected, only when it 
influenced by earth when both are on same sid 
of sun. Hence this correction in mandocca is t 
account for influence of earth. 

(c) Correction in sighrocca and mandocca 
Budha is to make correction of elliptic orbit 
Budha slghra (i.e. Budha itself) and its hi 
inclination with sun's ecliptic; 7° as seen from s 



True Planets 135 

These are reasonable assumptions of the 
origin. It needs further research and verification, 
gut obviously, these corrections tallied with obser- 
vations in author's time. 

Verses 90-103 - Manda and sighra Paridhi - 
For any planet, attraction by its mandocca is 
multiplied by kala of a circle (21,600) and divided 
> by radius (3438). Result is manda paridhi. Maxi- 
mum mandaphala varies with manda paridhi. (90). 

Sighra paridhi of mangala, guru and sani is 
more in even quadrants compared to odd quadrants 
end. For budha and sukra it is opposite. (91) 

According to Surya siddhanta - Manda paridhi 
of ravi is 14° in even quadrant and 13'4Q' in odd 
end. Similarly, mandapraidhi of candra is 32* in 
even quadrant and 20' less in odd quadrant i.e. 
3r40\ (92) 

According to Siddhanta Siromani of 
Bhaskaracarya, mandaparidhis of ravi and candra 
are constant and are 13 e 40 / and 31° 16' respectively. 
(93) 

I (author) have calculated the values of manda 
paridhis of ravi and candra by observing conjunc- 
tion of moon with stars and difference in rasi of 
moon and sun (phases of moon) (94) 

In odd quadrants, mandaparidhi of sun is 
12° 6' and of candra is 31° 30' (95) 

If manda kendra is at end of 4th quadrant, 
mandaparidhi of ravi is 11*30'. Now, method for 
finding manda paridhi at other places is being told 
(96) 



196 



SiddMnta Darpana 

-- - . * 

Multiply kotijya of ravi manda kendra by 6 
and dKde by radius (3438). Result wffl be in kala 
ete If manda kendra is in 1st or 4th quadrant, 
tSJLt this result from 18' ^ At other 
positions this will be added to 18 kala. (97) 

Either of the results is multiplied I by R cos of 

^Snd^Idhi afendtf od^adrant (IT*. 

wSn mandakendra is in 2nd or 4th quadrant. In 

other quadrants it is added (98) 

This method is adopted for accurate calcuk- 

tion For rough work, l/9th part of previous result 

Padded. This will give accurate results only on 

parva sandhi (Purnima or amavasya). 

Kotijya of candra manda kendra (R cosine) » 
multiplied by 30 and divided by radius (3438). 
SSttaK* etc is added to the manda paridh 
afend of odd quadrants &'*>>.***,££ 
kendra is in six rails starting from karka. In other 
PosSns, it is subtracted to find sphuta manda 

P Manda paridMs of planets are -mangala 69- 
budha 27% guru 34*30' sukra 12 and sam 39 
Sighra paridhis at end of even and odd 

quadrant ** tant 

237* 
Mangala 238' 

Budha 139° 

Guru 70 ^ 

Sukra 26 1 * 

Sani 39- * 



True Planets 197 

Notes (1) Mandaparidhi is correction method 
for elliptical orbit which have been assumed circular 
for first approximation. 

(2) Sun and moon are directly in an ellipse 
around earth. But other planets take their position 
aS a result of two orbits ~ orbit of sun around 
earth (apparent) and orbit of planet round sun. 
Correction from mean position due to smaller of 
these orbits is done through sighra paridhi. 

(3) The method of correction by manda and 
sighra paridhis will be explained after calculation 
of these for tara grahas. 

Verses 104 to 112 - Sphuta manda and sighra 
paridhis for tara grahas - Difference (of 1") between 
sighra paridhis at the end of odd and even 
quadrants is multiplied by bhuja jya of sighra 
kendra and divided by radius (3438) i.e. multiplied 
by sine of sighra kendra. 

This result is added to the sighra paridhi at 
end of previous quadrant, if it is rising in current 
quadrant. Otherwise/ it is substracted. 

For more accurate value of sphuta sighra 
paridhi of mangala, we add 1/30 part of bhuja kala 
of sighra kendra (i.e. R sine of sighra kendra is 

minutes). 

Manda paridhi of mangala is 69° only at the 
end of quadrant. To find the intermediate values, 
we select the lesser part of manda kendra - among 
lapsed part and remaining part in the quadrant. R 
sine of that angle is multiplied by 8" when manda 
kendra is in six rasis starting from karka, or by 4° 
w hen manda kendra is in 6 rasis starting from 
^akara. Result is divided by R sine of 1-1/2 *rasi 



19S 



Darpana 



(45°) = 2431. .Result is converted to degrees etc 
and added to 69° which gives sphuta manda 
paridhi of mangala (at any place). 

When manda kendra of mangala is between 
4868' and (4868' + 1590') or between (15,142') and 
(15,142'+1590'), its mandaparidhi is taken as equal 
to its mandaphala of 3 rasi i.e. 11 2'47". 

We find the lesser of lapsed and remaining^ 
parts in quadrant of Budha manda kendra. Its R 
sine (jya) is divided by 9 and result is substracted 
from manda paridhi (27°). We get sphuta manda 
paridhi. 

R sine of sukra manda kendra is multiplied 
by 2 and divided by radius R = 3438. Result in- 
degrees is substracted from manda paridhi (12°) to 
find its sphuta value. 

Notes - (1) As first approximation, planetary 
orbit is considered circle with earth at centre (for 
moon and sun). 




M 
:i 



Figure 14 - Epicycle (Niocca vrtta) 



*\ 



'■ 'A 

-A 






True Planets 



199 




Figure 15 • Eccentric Circle 



But orbit is elliptical with earth at focus which 
is away from centre at a distance of ae towards 
apogee (mandocca). Here a = semi major axis and 
e is eccentricity of ellipse. 

At first step of approximation, we shift the 
circular orbit by distance ae in direction of major 
axis (Fig. 15). YU X N is orbit of mean planet 
whose apogee is U and V is the point of mesa 0°, 
vernal equinox, from where longitude is measured. 
Another circle with centre at C in direction EU is 
drawn with same radius. In both the circles the 
planet rotates with same speed. Thus every point 
on the eccentric orbit (Prati vrtta) PUi L is at a 
distance CE in direction of U. At apogee, mean 
planet is at U and true planet at Ui. When mean 
planet is at M, corresponding planet on prativrtta 
is at Ti where MTi = EC and both are parallel. 

Same displacement can be done by assuming 
movement of spasta graha (true planet) on another 



200 Siddhanta Darpana 

small circle whose centre is on madhya graha. The 
circle is called manda paridhi (epiciycle) which 
rotates fixed with radius vector of mean planet. 
However, movement of spasta graha on manda 
paridhi is in opposite direction to the motion of 
madhya graha but with equal angular speed. At 
apogee position in Fig 14, mean graha is at U and 
true graha is at Ui in same direction, When mean 
planet moves to M in anti clockwise direction by 
angle 6 = Z.UEM (manda kendra), the true planet 
moves by same angle = TjBBi in opposite 
direction. Thus Ti is always in direction of 
mandocca i.e. MTi is parallel and equal to CE. 
Thus by construction of manda-paridhi also, all 
points of madhya graha orbit are shifted by distance 
EC in direction of EU towards ucca. Thus both 
the constructions are equivalent. 

(2) Ellipse is symmetrical with respect to centre 
but not from focus which is centre of true orbit. 
Next step of approximation to make it toally 
equivalent to elliptical orbit is by changing the 
radius (or equivalently circumference) of manda 
paridhi at different places. 

Let E be origin and EU direction of X axis, 
EX being direction of y axis. Radius of mean orbit 
(deferent) EM = R 

Radius of manda paridhi for manda kendra 9 
= m + n cos 6 

n has lowest value at 90° or 270" when cos 
6 = 

m+n has highest value at apogee (0=0 e ) or at 
6 = 180° (-ve) Coordinates of point Ti are - 

x = EM, cos $ + MTi - - in direction of EU 



-:« 



True Planets 201 

= R cos + (m+n cos 6) 

or x - m = (R+n) cos 6 - a cos 

y = Rsin0 = bsin0 

where a = R+n, b = R are the semi major 
and semi minor axis. This is parametric equation 
of ellipse with centre at (m,o) i.e. at distance EC=m 
from centre of kaksa vrtta towards mandocca. 

From this, 

e - V^P = V(R+n) z -R* = V 2T1 R + r, 2 
e - va — _d_ R + n R + n 



a 



360* 20* 1° , . ..-.■ , 

R = ^— , n = — = (surya siddhanta) 

2 Jt 2 jt 3x2^ 

Hence 



V -.360 + ^ 

^ ?_ = V & x 360 + 1 

e 360 + V^ 1081 

= 0.043 (real value for sun is 0.0167) 

Geometric equivalent of the correction - 

Without continuous varying the radius of 
mandaparidhi also, we can obtain the true position 
of planet. 

In figure 14, join CT, which cuts deferent 
(orbit of mean planet) at S. Produce ES and MTi 
to meet at T. Then MT is the radius of true epicycle 
at M and T is true position of sun. 

Similar construction can be made for prati- 
vrtta (eccentric circle) also in figure 15. CTi cuts 
deferent at S and ES cuts MTi produced at T which 
is true position of planet. 

(3) Bhujaphala is equal to equation of centre- 
In figure 14, Sun's mean longitude = arc VUM 



202 Siddhdnta Darparut 

True longitude = arc VUS 

Difference between the two, i.e. arc SM, is 
the sun's equation of centre 

MA is perp. to EU and T1B1 and SB be 
perpendiculars to EM or EM produced. T1B1 is 
called bhuja phala or bahuphala and BiM is called 
kotiphala. 

A 5 BiMTi and MAE are similar. Then 

Ti Bi .MA 
Ti M ■ ~ EM 

or TiBi, i.e. sun's bhujaphala 
TiMxMA 

EM 

14° 

TiM = radius of epicycle (mean) = - — 

MA = R Sin 0, EM=R 

Hence, Bahuphala 

14° 

Sin 6 



2 Jt 

14 X 6C 






Sin© 



In 
= 133/7 Sin = 0.388 sin 6 radians. 

With mean value of manda paridhi 11 "48', it 
is 0.327 sin 6 which compares well with the modern 
value of 0.334 SinO. Plotemy had given 0.416 sin 
6 radians. 

(4) In eccentric circle, geometric construction 
gives the method of successive approximation 
described later on while dealing with true speed. 

(5) Heliocentric amomaly through Sighra 
kendra. —Position of tara grhaas depends on two 
orbits - apparent orbit of sun round earth and orbit 



203 



True Planets 

f planet around sun. Smaller of the orbits is called 
lighra paridhi and madhya graha corresponding to 
average motion is bigger orbit. 





Figure 16 



-A' 



Figure 17 




Figure 18 

Like correction of elliptic orbit through manda t 
paridhi, correction from heliocentric to geocentric 
position is done through sighra paridhi. It can be 
an epicycle, or eccentric circle, as shown in figure 
18. 



204 Siddhanta Darpana 

It is to be proved that sighra kendra is same 
as heliocentric anomady and sighra phala is 
conversion from heliocentric to geocentric position. 
This will show correctness of sighraparidhi method. 

Figures 16 and 17 show the anomalies with 
sun as centre. First figure is for inner planets venus 
and mercury, marked as V. Second figure is for 
superior planets mars, jupiter and saturn, marked 
as J. 

Figure 18 is for sighra paridhi - both for 
inferior and superior planets. 

Figure 16 and 17 - S = Sun, E = Earth, SA 
= direction to mesa 0°, EA' = direction to 0°. SV, 
EV are heliocentric and geocentric directions of 
sighrocca. V is actual planet and V is imaginary 
point (sighrocca of inferior planets) Draw EJ' I I 
SJ. Radius of inner and outer circles are r and R. 
K is radius vector to the planet (from earth). 

Figure 18 - Ei = Earth's centre, E 2 - Centre 
of eccentric circle (It will be proved as centre of 
sun) Mi, M2 are mean planets in deferent (kaksa 
vrtta) and eccentric (Prati Vrtta). E a E 2 * r = antya 
phala jya. R is radius of both circles. K is radius 
vector to planet known as sighra karna. 

Sighra kendra (anomaly) = Longiude of 
sighrocca - Longitude of planet (fig. 18) 

= L aEi Ai - Za E1M1 = Lb! E 2 Ai - Za' E 2 
M 2 = m 

In Fig 16, ZA' EVi is longitude of sighrocca, 
ZA'ES is longitude of sun treated as madhya graha 
of inferior planet. 

sighra anomaly, m = ZVSS' = ZV'ES 



True Planets 205 

= ZA'EV - ZA'ES 

In fig. 17, m = L S' SJ=/L SEJ = L 
A'ES-iiA'EJ' 

= L A'ES-ii ASJ 

= Longitude of Sun (treated as sighrocca of 
superior planet) - longitude of planet from Sun 
known as mandasphuta graha. 

= Sighra kendra 

Consider A s ESV, JSE and E1M1M2 of the three 
figures. 

L ESV = 2LJSE = L Ei M1M2 = 180* - m 
If value of sighra paridhi is taken such that 

SV SE M1M2 

— = — = (1) 

SE SJ Ei Mi K) 

all the three triangles will be similar. 

Thus sighra kendra is same as heliocentric 
anomaly and sighraphala L MiE M2 = L SEV = 
L SJE. 

K 2 = R 2 +r* + 2 Rr cos m (2) 

Comparison of values of orbit known in 
modern astronomy , shows that value of sighra 
paridhis have been chosen correctly, so that 
equation (1) holds - 

. Sighra paridhi small orbit (radius or circum) 

— ' 360° "~ Larger orbit ' 

Hanet Sighra Deferent Ratio Value in*modern 

paridhi astronomy 

(average) earth = 1 



Mercury 


139.5 


360 


.3875 


0.387 


Venus 


261.5 


360 


.726 


0.723 


Mars 


237.5 


360 


1.519 


1.52 



206 Siddhanta Darpam 

Jupiter 69.5 360 5.18 5.20 

Saturn 38.5 360 9.35 9.5 

(6) Formula for sphuta paridhi - Difference 
between values at end of odd and even quadrants 

is 1 . 

Hence if is angular difference from lowest 
position, addition will be Sin 6 in degrees. 

For further correction in mangala 1/30 part is 

. 31 sin 6 --rt-oo o- a 
added i.e. correction is = 1.033 bin v 

Mangala manda paridhi is minimum in ends 
of quadrants (90° interval) and it is maximum in 
between (45° from ends). Difference from minimum 
69° is 8° in 90° to 270" and 4° in other half. Hence 
correction in 90° to 270" fe- 

8° x sin 6 4° sin 6 

+ in other half it is r— ; — — 

R sin 45° R sin 45° 

It is constant in two intervals 4868' and 4868' 
+ 1590' and (15,142 to 15,142' + 1590'). Then 
mandaparidhi is 11°2'47'. 

Budha manda kendra = lesser interval from 
ends of quadrant. 

R' 
Sphuta paridhi = 27° - — sin $ 

Sukra manda kendra = 

Sphuta manda paridhi 

= 12° - 2° sin 6 

(7) For outer planets, earth is on same side 
of sun and closer to # planet for sighra kendra ift 
even quadrants (closest at end). So its sighra 
paridhi is more. For inner planets, it is opposite. 



,3 

.1 



True Planets 207 

Minute chanes in slghra paridhi are due to 
eccentricity of slghra orbit also. 

(8) Bhaskara II, has measured difference of 
mandocca - madhya graha in anti clockwise 
(position direction) and madhya - sighrocca in 
opposite direction, Madhya is faster than mandocca 
but slower than sighrocca. However, both 
measured same way make no difference. 

Verses 113 to 120 - Bhuja and koti phala and 
karna 

According to surya siddhanta, sphuta manda 
paridhi multiplied by R sin of bhuja of manda 
kendra and divided by 360* gives bhuja phala. 
When this paridhi is multiplied by kotijya (R cosine) 
of bhuja and divided by 360° it gives kotiphala. 
(113) 

When arc is smaller than 225', it is same as 
its R sine (jya). Then arc or R sine need not be 
converted to each other. (They are taken equal). 
Only when arc is more than 225', its sine is to be 
calculated. 

According to surya siddhanta, slghra kotiphala 
is added to trijya (3438) when slghra kendra is in 
six rasis beginning with makara (i.e. 270° to 90°). 
For other sighra kendras (i.e. 90° to 270°) it is 
substracted from trijya. 

This is kotija bhujaphala, used for correction 
°f radius and should not be considered an arc. 

In surya siddhanta - Squares of bhuja and 
koti phala are added and square root of sum is 
taken. Then we get slghra karna. Bhuja phala 
Multiplied by trijya (3438) and divided by slghra 
karna gives sighraphala in minutes of arc. Slghra 



208 Siddhattta Darpatui 

phala is used for first and fourth corrections o| 
five star planets starting from manga la. Sun and 
moon become spasta with only one correction with 
mandaphala. But in five tara grahas, sighra phala 
correction is done at first, then mandaphali 
correction is done twice. At fourth step, sighra 
phala correction is done again. When sighra kendra 
or manda kendra is less than 6 rasi, sighra of 
manda phala is positive, hence always added for 
correction. When kendra is more than 6 rasi, phala 
is substracted. 

Notes : (1) List of given formulas 

Manda Paridhi __ Mandatrijya 

360° " Trijya(3438) 

Hence manda trijya r, bhuja of manda kendra 
give 

m, • u i r • R sin ° • a 

Bhujapnala = — = r sin 6 

In sighra paridhi, R + r cos $ is calculated 
for known distance of true planet. Cos 6 is positive 
from 270° to 90° hence it is added, otherwise 
substracted. 

Stghra Kama = Vfihuja Phala 2 + Kotiphala 2 

(both o f sighra paridhi) 

i.e. r * Vj* s j n 2 q + j2 cos 2 £ = radius of sighra paridhi 

or sighra karna, 6 = bhuja of sighra kendra. 

c- u u i • a r • R sin 
Sighraphala = r sin 6 - 

Proofs are obvious when diagram of sighra 
or mandaparidhi is seen. Sin is positive when B 
is between 0° to i80° hence manda or sighra phal* 
is positive and is added. 



True Planets 209 

Verses 121 - 123 - Correction in madhya tara 

graha — Madhya graha corrected by half of sighra 
phala (addition or substraction) gives first graha 
(corrected). Manda kendra is calculated for first 
eraha and half of its mandaphala correction gives 
second graha. Manda kendra is again calculated 
for second graha. Its correction by mandaphala 
(full) gives third graha. For third graha, sighra 
kendra and sighra phala is calculated. On correction 
of third by this sighraphala, we get fourth graha 
which is the true position of planet (spasta graha). 

Notes (1) - If madhya graha is Po^ 1st and 
4th sighraphala are Si, and S4, 2nd and 3rd 
mandaphala are M2 and M3, graha after 1st, 2nd, 
3rd, 4th correction are Pi, P 2 , P3, P4 then 

Pi = Po ± J 

M 2 
P 2 = Pi.±Y i 

P 3 = P 2 ± M3 
P4 = P 3 ± S4 

Here Si and S4 are calculated for Po and P3, 
and M 2 and M3 are calculated for Pi and P 2 . 

Correction order can be indicated by 
S M w ■ 

- + y + M + s 

(2) Aryabhata method For superior planets 

2 + y + M + s 

For inferior planets (vernus and mercury) 



220 Siddhanta Darpana 

S 

- + M + S (only 3 steps) 

He has calculated slghra kendra in opposite 
direction (Sighrocca - planet), hence it is substracted 
for 0° to 180°. 

M S 
Bhaskara I method — + — + M + S 

2 2 

S 
For inferior planets — + M + S (only 3 steps) 

t 

S/2 for inferior planets is corrected in reverse 
way and sfghra kendra is calculated from its 
mandocca. 

Surya siddhanta method is the traditional and 
most popular method in country. It has been! 
followed in siddhanta darpana also. 1 

(3) Further explaination of variations in manda ^ 
and slghra paridhi. (In continuation of note 2 after j 
verse 112) I 

772 - 

Eccentricity e of orbit = — I 

a i 

where m smallest value of manda paridhi 1 

a = semi major axis = R+n I 

I 
n = difference between maximum and mini^ 

mum values of manda paridhi. 

Thus e = :r = — approx, n very small. 

R + n R rr J 

It is also given by e = V2r ^ R + n2 

R + n 

* V r 2ft approx as n is very small. 
R 



True Planets 211 

Thus approximately, e = — = V^n 

R R 

or m = V2«R 

m 
or n = — , m = eR 

This gives method of calculating maximum 

manda pardhi and its correction term For sun, 

max. paridhi is 14° (= 2 Jim) and max correction 

is 20' (= 2#n) 

e = v^ = 0.043 
R 

m 
e = 039 

K 

This is similar by both method. Thus correc- 
tion depends on value of max manda paridhi. 

(4) Reasons for starting correction with sighra 
phala - Mandaparidhi is measure of eccentricity of 
orbit (e=m/R) which is very small and less than 
1/50. Shighra paridhi is ratio of smaller orbit to 
bigger orbit among the orbits of earth and planet 
round the sun. This varies from 1/9 to 3/4 
approximately. Hence at first step manda correction 
can be neglected and only sighra correction in 
done. 

For inferior planets manda correction also is 
done in sun's orbit, not in the orbit of planets. 
Hence alternatively, manda correction can be done 
before sighra as stated by Aryabhata and Bhaskaral. 

We do not calculate manda or sighra kendra 
from true planet, but from mean planet which is 
an approximation. Hence only half corrections are 
done for sighra and manda in beginning. Prob- 



212 Siddhdnta Darpatut 

ability of negative or positive error will be both 
equal in half corrections and are likely to cancel 
each other. Then manda correction in full gives 
heliocentric anomaly of the planet - called manda 
spasta graha. Its last correction by sighra phala has 
been explained in note (3) after verse 112. 

Since sighra and manda corrections are com- 
parable, their half correction only is taken at a 
time. After 2nd correction error is reduced and 
after full manda correction, exact sighra phala can 
be determined. 

(5) This is a type of calculation based on 
probabilistic value of errors which is called 'Monte- 
Carlo method' in modern numerical analysis. 
Reduction of error at each step is similar to 
'iteration method' for system of non-linear equa- 
tions. 




"o Xt Xg X 

Figure 19 - O<0*<1 



*» X 2 X x t 

Figure 20 for - 1 <0'<O 



Method of iteration for numerical solution. 
Solution for y = <D(x) is its point of intersection 



711 

True Planets 

with line y = x whose slope with x axis is 1. Figure 
19 explains the approximations when slope is 
positive and figure 20 indicates negative slope of 

y = 4>(x) Slope is <*>' (x) or — , it is positive when 

function in increasing, negative when it is decreas- 
ing. In both cases its numerical value is less than 
1 i.e. slope of y= x. Only in such case successive 
approximations will reduce the errors at each step. 
For slghra and manda corrections also, the correc- 
tions are much smaller than 1 as explained in 
previous para. 

Xo is the first approximation (like madhya 

graha). x a , x 2 , x 3 are next approximations. 

When function (slghra phala or manda phala) is 
increasing, i.e. correction is additive, all the 
approximations are on left side of, or less than 
true value x. When correction is negative, i.e. 
function is decreasing (Fig. 10) xi, x 2 , x 3 - - - 
alternate on either side of the true value. 

In both cases, diagram shows that errors 
decrease at each step, which was purpose of our 
corrections. 

(6) Reasons of half corrections in first two 
steps - By full correction we may over correct and 
may not decrease the error which is required for 
iteration. Half correction will always reduce the 
error. Full investigation can be done only # with 
Lyapunov's conditions of stability. However taking 
half of the approx value of correction, probability 
of positive and negative error both are same and 
it wil be approaching zero in end. 



214 Siddhanta Darpana 

It is. similar to methods used by computer 
which divide the line segment into two parts for 



Y 




X. b s b-bo 



Figure 21 

numerical approximation. In figure 21, solution of 
f (x) = is its intersection with x axis. On one 
side of true x*, there is a point a Q for which f(x) 
is - ve and on other side f(b c ) is + ve. We take 
midpoint c of interval (a G , b ). If f (c) is negative, 
we make it the new point in place of a, where it 
is negative. Thus we go on dividing the interval 
for better approximations. 

Verses 124-131 - Special correction for 
mangala and budha - For mangala, 3rd and 4th 
phalas in kala are muthplied and divided by 10. 
Result is substracted from last karna. 
Then we get result in lipta etc. This result is added 
to 4th (sphuta) graha when manda kendra is 0" 
to 180 °, otherwise substracted. If 3rd kendra 
(manda) of mangala is from 90° to 270°, 4th phala 
is substracted from 55, result is multiplied by 
manda koti phala of 3rd operation. This result is 
substracted from 4th karna. This is substracted from 
5th graha, then we get 6th sphuta graha. If manda 
kendra of 3rd planet is 270° 'to 90° then this 
correction is unnecessary, (fifth graha will be true). 

Madhyama budha is substracted from budha 
sfghrocca, already corrected for parocca* This slghra 
kendra is used to find half of first slghra phala, 



True Planets 215 

which is kept in 1st place. In 2nd and 3rd places, 
we keep half mandaphala obtained from madhyama 
budha after 2nd operation (correction with half 
manda phala). 

At 2nd place, this mandaphala is multiplied 
by half sighra phala at 1st place and divided by 
half of the 4th sighra phala. 1/3 of the result is 
substracted from mandaphala at 3rd place. 

The new manda phala is used to make 3rd 
correction of madhyama budha. From that 4th 
sighra phala is obtained and kept in 2 places. At 
2nd place it is multiplied by 3rd kotiphala divided 
by radius (3438)). Result is added or substracted 
at 1st place (addition is done when manda kendra 
is 90° to 270°). This sighraphala is used for 4th 
correction. Then we get more correct result com- 
pared to Surya siddhanta. 

Notes : The rules are lengthy and confusing 
when stated in words. 

(1) Rules for mangala - 

Po, Pi, P2, P3, P4 are the mean planet and the 
planets after 1st, 2nd, 3rd and 4th correction. Si, 
S 4 are sighra phala for 1st and 4th corrections, 
when sighra kendra is calculated for Po and P3. 
M 2 , M 3 are mandaphala for 2nd and 3rd correction 
where mandaphala is calculated from manda 
kendra of Pi and P 2 . # 

Thus Pi = Po + Si/2 (S & M may be + ve or 

- ve) 

M2 
P2 = Pi + — , P3 « P2 + M3, P4 = P3 + S 4 

^ Si Mz „ „ . 

Thus P4 = Po + -z- + -r- + M3 + S4 = True graha 



r4 __ — — J n liptag 



216 Siddhanta Darpana 

r\, r 4 are sighra radius for Si, S4 and ri, r 3 
manda radius for M2, M3. If is manda or Sighra 
kendra (bhuja), 

M or S = r Sin 6 For mangala we obtain P 5 

and Pe and further corrections of true planet. 

f M3XS4 x 

Fifth correction X5 = 

P5 = P4 + X5 when manda kendra of P4 is 
between 0° to 180° 

or = P 4 - x 5 when it is between 180° to 360°. 

When manda kendra of P3 is between 270° 
to 90° this P5 is the last correction needed. If 
manda kendra of P3 is between 90° to 270 - , 

sixth correction 

xe = r 4 - (55 - S 4 ) r 3 cos 3 

P 6 = Ps - xe 

(2) Correction for Budha - S' is sighra phala 

Si 
of budha corrected for parocca. From — we 

calculate M2'. For third correction we do not 
calculate Nfe' from 2nd planet. 

/ c ' \ 

1- S1 






3rd correction = M2 



= X 3 



3S 4 

S 4 is calculated by general method. 

The new sighra phala after P 2 + x 3 = F3, i&| 
called S 4 '. 

Fourth correction x± = S 4 ' (1 ± r 3 Cos ^3) 

Addition is done when 3 is 90° to 270° 

P 4 ' = P 3 ' ± xa (addition for 0° to 180°) 



True Planets 21? 

P 3 ' and P4' are planets obtained by revised 
method. 

Verses 132-138 - True speeds of sun and 

moon. 

Now true speed of graha is considered. The 
speed changes every moment, but sphuta gati of 
a day is the difference between sphuta graha on 
two successive days. Strictly this will be average 
daily speed for that day. 

Dainika gati of mandocca, substracted from 
dainika gati of mean graha, gives danika gati of 
manda kendra. Dainika gati of manda kendra 
multiplied by *manda kotiphala and divided by 
radius gives manda gatiphala for one day. This is 
added for manda kendra in 6 rasis from karka ; 
in madhya gati of graha. Other wise it is 
substracted from madhya gati. This result will be 
manda sphuta gati for one day i.e. from sunrise 
to the next sunrise. 

At sunrise, difference of true moon and true 
sun gives the balance part of current tithi. This 
(added to sunrise time) gives ending time of tithi. 
At that time true moon is again calculated and 
further correction of tithi end time is done. This 
accuracy in knowing beginning and end of tithi is 
needed only for ascertaining time of eclipse or oi 
sraddha (last rites). For normal works, the # true 
position of sun and moon and their speeds al 
sunrise will be assumed constant for the day. 

Notes : (1) List of all terms as revision and 
summary - 

Mandocca-Madhya graha = manda kendra M. 



218 



Siddhanta Darpana 

* 

Sighrocca - madhya graha = Sighra kendra S 
Manda kendra or Sighra Kendra = 6 




Figure 24 

True position of graha is P' while mean graha 
is at P. Radius r of manda or sighra paridhi is 
PP. 

EP' is karna = K (Sighra or manda) 

EP = R, radius taken a§ 3438'. 

2 n r is expressed in degrees of manda or 
sighra paridhi. 

PN' is perpendicular on Karna EP', N' is true 
position on Kranti Vrtta. Thus PN' is the correction 
in mean motion called sighra or manda phala. 

Mandaphala = PN' = almost P'N. 

It is slightly less than P'N, perpendicular from 
P' to EP extended. 

P'N = Doh phala or Doh jya 
= r Sin 

w j . i t>v„ r sin x R 

Mandaphala PN =— 

r R + r cos 6 

Kotiphala PN is addition to the mean trijya 
in that direction PN == r cos 6 



Sighra or manda karna K = EP' 



■1 



True Planets 219 

K 2 = (R + r cos 6) 1 + (r sin Of 

= R 2 + r 2 + 2 r R cos 6 

(2) Now the speed can be calculated with help 
of differential calculus. These results cannot come 
by any other method and are according to surya 
siddhanta. 

In figure 24 in above para, <S> is angle of ucca 
point U with mesa 0° at A. Then madhyagraha at 
P = <£ + 0, mandocca = 

dP dO d 

Thus d7 = ^r + IT (t 1S tune) 

or dainika gati of madhya graha = gati of 
mandocca + gati of manda kendra. 

True graha is at N' = P + r sin (negative 
correction) 

dN' dP a d0 

or = — + r cos — as r is constant. 

dt dt at 

Thus additional gati i.e. correction == r cos 



.e, 



d6 



dt 
= kotiphala x gati of manda kendra. 

Verses 139-142 - Sighragati of tara graha. 

Sighrocca gati —madhya graha gati = Sighra 
kendra gati. Sighra phala is substracted from 90 °, 
it is multiplied by daily motion of sighra kendra 
and divided by sighra karna. Result substracted 
from sighrocca gati is sighra sphuta gati. If it is 
nu>re than sighrocca gati, reverse substraction gives 
retrograde motion. In this way .5 tara graha have 
two types of gati — manda sphuta and sighra 



220 Siddhanta Darpana 

sphuta. Ravi and candra have only manda sphuta 
gati. 

Notes : (1) Like above, madhya graha (manda 
sphuta for sighra gati) is at P, and P' is sphuta 
graha. 

Sphuta kendra O-P is given by O where O 
is longitude of sighrocca. 

dO - dP dO 

"77 = j. + "TT/ t is time measured in days 

i.e. dainika gati of sighra kendra = gati of 
sighrocca - gati of madhya graha 

P is sphuta, its component perp to radius is 
4 P + r sin $ 

dP' dP n d0 

Hence -3- = — + r cos — 

at at at 

dO d$ „ d$ 

= —rr - "TT + r cos0 . t~ 
dt dt dt 

dO d0 

= dT"& (1 " rc09fl > 



dP' d O d 



( R- Rr Cos \ 
R 



dt A d * 
Thus negative sighra gati phala is 
R - r . R cos 
R 

sin 90° - Kotiphala of sighra 

sighra Karna v PP / 

(2) Exact derivation assuming variation erf 
karna also - 



True Planets 



221 





Figure 25 Figure 26 

(Inferior planet like venus, mercury) superior planet 

E = Earth, S = Sun, V= inferior planet (Fig 

25) 

J = Superior planet (Fig 26) 

SA, EA' = direction of mesa 0° from sun .and 
earth 

m == sighra amomaly, n = sphuta kendra 

R = bigger orbit radius = SE in Fig 25 

or SJ in fig 26. 

r = smaller orbit radius = S V in fig 25 

or SE in fig 26. 

K = Sighra karna i.e. distance from earth to 
planet (true) = EV or EJ. 

True motion of planets <5 (A'EV) or 6 (A'EJ) 

But 6 (A'EV) = d (A'EV' - n), and 6 (A'EJ) 
= d (A'ES-n) 

For inferior planet, d (A'EV) = 6 (ASV) = 
sighrocca gati, d n = sphuta kendra gati. 

For superior planet, <3 (A'ES) = Sighrocca gati 
(sun is sighrocca for superior planet) 

(3 n = sphuta kendragati as before 

Thus in both cases, 



I 



222 Siddhdnta Darpana 

Sphuta gati = slghra gati - sphuta kendra | 
gati- (1) | 

To find <5 n, from figures 1 

K cos n - R cos m = r (2) j 

Differentiating (2), we have 

- k sin n d n + cos n. <5k + R sin m d m = 
(3) 

But k 2 = R 2 + r 2 + 2 R r cos m 

Differentiating, 2K6 K = - 2Rr sin m <5m - (4) 

Eliminating (5k between (3) and (4) 

R r sin m d m x cosn 
— k sin non- — + R sin m S m= 

Jv 

r 
or K sin n 6 n = R sin m d m (1 - - cos n) 

k 

R sin m dm 
= (K - r cosn) 

But K-r cos n ,- R cos E 

R sin m dm x R cos E 
So on = 

K sin n 

But R sin m - k sin n 

R cos E dm 
so <5n = — 

(3) Proof of approximate method. 

<5 m x R 
Mandagati phala = - — — 

dm - mean motion, k - manda karna 

d m x R 



Thus mandagati phala = 



V R 2 + r 2 ±2Rr 



cosm 



2r -1/2 



= d m (1 ± — cos m) " neglecting square of -r 



_ r 

~ dm (1 ± — cos m) 

R 



.■j 



True Planets 223 

r R cosm <3m 



5 <5m± 



R * R 
Kotiphala x manda kendragati 



-&n± . R 



(4) Approximation of true distance and daily 
motion can be done by epicyciic or eccentric circles 
also by successive approximation. That can be seen 
in commentary on Mahabhaskariya by Prof. K.S. 
Shukla published by Lucknow University. 
Geometric explanation of Lalla method can be seen 
in commentary on Sisyadhivrddida tantra by Smt. 
Bina chatterjee published by INSA, Delhi-2. 

Verses 143-150 - Gati phala at four stages for 
tara grahas- When sighra sphuta gati is more than 
daily mean motion, then madhyama gati is sub- 
stracted. When daily mean motion is more, sighra 
sphuta is substracted from it. 

When sighra sphuta gati is vakra, it is added 
in daily mean motion. The result in either of three 
cases is called first gati phala whose half is taken. 

When sphuta sighra gati is more, it is added 
in daily mean motion. When sphuta sighra gati is 
less or vakra, 1st gati phala is substracted (half 
only) from madhya gati. This is 1st corrected gati. 

First gati is forward or reverse. It is multiplied 
by manda kotiphala and divided by trijya. Half of 
the result (2nd gati phala manda) is taken. 

When manda kendra is in 6 rasis starting from 
karka (90° to 270 °), 2nd gatiphala half is added to 
1st gati otherwise substracted from 1st gati. Result 
will be 2nd gati. 

Again 2nd gati is multiplied by manda 
kotiphala and divided by trijya (3438) and full result 



224 Siddhdnta Darpang 

(3rd gatiphala) is used for correcting 2nd gati 
(addition or substr action). Result is 3rd gati. 

Third gati is multiplied by slghra koti phala 
and divided by slghra karna. Result is 4th gati 
phala, which is used to correct 3rd gati. Then 4th 
gati is the true gati. If 4th gati phala is more than 
slghrocca gati then motion is reverse (vakri gati). 
This method is more correct then surya siddhanta. 

Verses 151 to 158 - Special methods for true 
speed- When mandocca of mangala, budha or sard 
is moving forward, its speed is substracted from 
1st and 2nd gati. If it is vakri (in reverse motion), 
its speed is added. From these corrected, 1st and 
2nd gatis, we find 2nd and 3rd gati. New 2nd and 
3rd gati are substracted from sphuta slghra gati for 
1st and 4th gati. 

Fourth gati phala of budha is kept in two 
places. At one place, it is multiplied by manda 
kotiphala and divided by radius (3438). Result is 
added to 4th gati phala in second place, when 
manda kendra is in six rasis starting from karka 
(90° to 270°). Then sphuta gati of budha will be 
more correct. 

If Sighra gati half is vakra, it is added to 
negative mandagati half or difference is taken from 
positive half mandagati: Result will be vakra 
(reverse) gati. Mandagati (2nd or 3rd steps) phalas 
are added in six rasis starting from karka and 
substracted otherwise. 

Many methods of finding true planet from 
mean planet are coming to mind, but these are 
not given here (by author), as they are very 
complicated. . 



225 



True Planets 

Daily true motion is used for finding transition 
time of graha from one rasi to next or in 
conjunction (yuddha) of planets. For ravi and 
candra, method is different. 

Notes : Method for sighragati phala and 
mandagati phale has already been explained. 
Reasons of vakra gati will be explained when its 
starting or ending point are calculated. 

Verses 159-160 - According to Surya siddhanta 
tara graha becomes vakri when its 4th sighra kendra 
has the given values - 

Mangala 163" Sukra 167° 

Budha 146* Sani 115" 

Guru 126° 

At 4th sighra kendra obtained by substracting 
these values from 360°, the graha again becomes 
margi. 

Notes (1) Derivation of 4th sighra kendra for 
vakri gati. We assume that heliocentric orbits of 
earth and Jupiter around sun are both circular and 
coplanar. 




« 



► u 



Figure 27 - Explanation of vakri gati and its position 



226 Siddhanta Darpana 

Let S »= Sun, E = Earth, J = Jupiter, u = 
earth's linear velocity. Sighra karna EJ = k, 
v= velocity of jupitar. 

r and R are orbital radii of earth and Jupiter. 

EE' and JJ' are parpendiculars to EJ so that 
when relative velocity of Jupiter with respect to 
earth, i.e. perp to EJ is zero, Jupiter will appear 
stationary as seen from earth. 

This means that u cos0 + v cos e - 

- (1) 

U - COS £ 

or — = — (2) 

v cos w 

From AESJ, R cos <p + k cos 6 = r - - - (3) 

r cos <p + k cos e = R (4) 

t, ,~* , ^w cose rcos0-R ^ 

From (3) and (4)), = \ ..(5) 

cos0 R cos <p - r ' 

COS £ 

Equating — from (2) and (5) 

u r cos <f> — R 



v R cos <p — r 

ru + Rv 

cos0 = r=— 

^ r v + Ru 



so that 



If m is sighra anomaly, then m = 180° - <p 

Y ru + Rv x 



So, cos m = - 



... (6) 



r v + Ru 

This is equivalent to formula given by 
Bhaskara II 

,_ , R cos E . dm 
spasta gan = sighra gati — 

i. «. .* ,- » R cos E . <5m .__ 

spasta gab - 0, if Sighra gab « — — : — - - (7) 



True Planets 227 

i.e. Jupiter appears stationary as seen from earth, if 

R cosE . 3m 
sighra gati = ^ 

u v 

Angular velocity of earth and Jupiter are - and — 

so that sun's apparent velocity is also u/r 
m == Kendra gati = sun's apparent velocity 
- Jupiters heliocentric velocity. 

u v_ 
= 7 " R 
Substituting this in (7), 

u Rcos E u v. 
sighra gati = - = g ( r - p) 

u E R cosE v_ 

.-.- (Rcos--l) = — ^— x R 

Here E = e , R cos E -K = - r cos0 

So - x (- r cos 0) = v cos E 
r 

or u cos + v cose = 

which comes to equation (1) 

Sighra kendra m is obtained from (6) 

where r = radius of sighra paridhi or antya 

phala jya, R = 3438', u = mean velocity of sun 

and v = mean velocity of the planet. 

(2) Some observations on aghra phala and 

sighra gatiphala - 

We have M 2 + E 2 = S ........(1) # 

Where M 2 = Mandasphuta graha, E 2 = Sighra 

phala, s=sphuta graha. 
Differentiating this 
<5M 2 + <5E 2 = <5S (2) 



22 # Siddhdnta Darpat 




i.e. Mandasphuta gati + sighra gatiphala | 
= spastagati | 

(a) Let E 2 be maximum so that (5E = 0, 
then (5M 2 = <5S 

This means that when sighra phala is maxi-l 
mum for sighra kendra 90° or 270° (Sin m 
maximum) manda sphuta gati is the spasta gati. : ] 

(b) Planets starts retrograde motion only after; 
the spasta gati vanishes i.e. <5M 2 + 6E 2 = 

Taking 6M 2 to be almost a constant, since^ 
mandagati phala is small, the negative value oft 
<5E 2 must cancel (5M 2 . (5E 2 becomes negative when 
sighra kendra is between 90° to 180° or 270° to 
360° when value of sine decreases in value. From! 
180° to 270° it is negative, hence its net vahiei 
increases. Thus the planet will have zero velocity" 
at two points symmetric to 180° (S' towards a and 

Thus if retrorade motion starts at 180° - it 
will stop at 180° + = 360° - (180 °-0), where its 
velocity becomes 0. 

Keeping earth constant, an inferior planet goes 
anticlockwise whereas superior planet goes clock- 
wise which is direction of sun's motion. 

(c) Values of spastagati at S 7 and C will be 
by putting R Cos E = R in formula) 

c * ^ 6- i_ ^ R cos E 6 m 
Spasta gati = Sighragati — 

Here, sighra gati = U, dm = U-V 

K = R + r at S' 
R - r at c 



True planets 229 

U - R (U - V) 
Spastagati = ^-^ at S and 

U - R (U " v ) 

RV-rU 

respectively. 




R + r R- r 

if R 2 V-2 RV + Rr U - r 2 U > R 2 V - Rr U+rRV 
_r 2 U 

i.e. if rR (U-V) > Rr (V-U) 

i.e. U-V is + ve and and equal to V-U. 

Thus positive velocity at S' of the planet will 
be equal to its negative or retrograde velocity at 
c. Thus velocities direct or retrograde will always 
be less then S or c at any point between them on 
either side. 

Verse 161-164 : Udaya (rising) and asta 
(setting) of planets is of two types - practical is 
rough (sthula) and drik siddha is suksma. (accurate) 

The planets set in west when their sighra 
kendras cross the following values - 

Mahgala 332° Sukra 177* 

Budha 159* Sani 343° 

Guru 346° 

For rising in the east, last sighra kendra is 

Mahgala 28° Sukra 183* 

Budha 201* Sani 17* 

Guru 14* 

For setting in east, sighra kendra of inferior 
planets are, Budha 310° Sukra 336° 



230 Siddhanta Darpana 

For rising in west, slghra kendra are 
Budha 50° and Sukra 24° 

Notes : (1) Rising of a planet means that it 
is above horizon of earth. But tara graha are visible 
only during night time, so their rising is only seen 
at night. 

Obstruction due to sunrays makes the tara 
grahas invisible during day. When they are away 
from sun sufficiently, they can be seen. That is 
called heliacal rising or drk siddha udaya. 

(2) Sun's velocity is greater than superior 
planets, so sun overtakes them so that they set in 
west and rise in the east. When these planets are 
situated within particular limits from the sun, they 
will be invisible in the rays of sun. Thus they will 
be invisible at conjunction with sun and within 
particulars limits from position of sun. The total 
difference from sun depneds not only on difference 
in longitudes, but also on difference in sara (north 
south distance.) 

The limits of invisible distance from sun 

depends on their distance from sun and relative 

brilliance. The brilliance also depends on their 

phase, i.e. part of illuminated disc facing earth. 

1 + cos EPS 
Phase is , ZEPS = sighraphala E2, 

1 + cos E2 
hence phase = 

At conjunction E2 = 0, entire planet will be 
iluminated but we cannot see them, because they 



True Planets 231 

^vill be immersed in rays of sun. With increase in 
E 2/ Cos E 2 will decrease and lesser part of disc will 
be illuminated. Since distance also will decline, 
luminousity will not be affected, (from S' to a). In 
path acb, planet gains in illumination and distance 
also decreases. Thus superior planets appear more 
and more brilliant when they are retrograding, 
being most brilliant at c. 

Spherical radius of jupiter, saturn and mars 
are in decreasing order, so that they will be visible 
at angular distances in increasing order. Inverse 
square law of reduction in brilliance with distance 
(karna) works but doesn't counter the effect of 
sizes. Thus sighra kendra of these planets are 14°, 
17°, 28°. In udayastadhikara, Kalamsa is slightly 
less, because distance will be (sighra kendra - 
sighra phala.) 

(3) Inferior planets rise heliacally in the east 
after inferior conjunction and then they are 
retrograde. They attain gradually the maximum 
elongation in the east, then direct motion starts. 
When elongation gradually decreases and after 
going ahead of sun, they set in east. Thereafter, 
they heliacally rise in the west. There again, their 
elongation attains a maximum value, after which 
they become retrograde. After crossing sun again 
they gradually set in west and rise in east. (Figure 
18 may be seen). 

When the sighra anomaly of budha and sukra 
are 50° and 24°, their sighra phala will be 13° and 
11°, so that they are the kalamsa i.e. elongation 
from there mean sun. Then, they rise in west, 
being near superior conjunction. When their sighra 
anomalies become 159° and 177°, same sighra phala 



2 ^2 Siddhanta Darpana 

will arise, so that they set heliacally in the west. 
Then as sighra kendra attains symmetrical values 
on other side of 180° i.e. (360°-159°) and (360M77*) 
i.e. 201° and 183 °, slghraphala are same, they rise 
in the east. Again, when they obtain sighra kendra 
(360°-50°) and (36°-24°) i.e. 310° and 336\ they 
set in the east due to same sighra phala or kalamsa. 
Verse 165 - Moon sets when it is 11° behind 
sun and rises again when it is 11° ahead of sun. 

Note : This is not related to rising in east or 
west. It is visibility near sun, which starts after 
11° distance from sun. 12° difference from sun 
makes 1 tithi (in 360° difference there are 30 tithis 
15 in bright half and 15 in dark half). Thus in 
amavasya, moon is not visible. It is again visible 
slightly before 2nd day of bright half (12° advance 
of sun). Thus start of 'duja' in muslim calender is 
counted from sighting of moon. 

Verse 166 : To find mean planet knowing the 
true. 

Assume the true planet to be the mean; 
compute the manda and sighra phala and apply 
them inversely. We have approximation of the 
mean planet. Treating this as mean planet, again 
obtain manda and sighra phala and apply them 
inversely. The process is repeated, till constant 
values are obtained. 

Notes : This is method of successive ap- 
proximation 

Verses 167-187 - Use of tables for calculation 
of true planets. 

Calculation of true planets is very long and 
difficult process and there are chances of error. 



:& 
.'j* 



True Planets 233 

Hence I (author), am giving correct Khandaphalas 
in a chart for easy calculation (167.) 

In appendix, there is chart of manda and 
sighra phala, for parts of to 24 (24 parts of a 
quadrant of 90° are 3° 45' each). This contains 
kotiphala of all planets, gatiphala of tara grahas, 
gatiphala of ravi and candra, sighra of 48 parts 
(180 °), difference of khandaphala, sighra karna in 
lipta (minutes of arc), degrees for cakra entry, 
kranti, sighra kendra for rising and setting etc 
(170). 

From the values in the chart, manda kendra 
bhujaphala in degrees, minutes, seconds etc are 
separately multiplied by 8, vikala (seconds) etc. are 
divided by 60, when they become degree, they are 
added to the degrees. Total degrees are divided 
by 30 to make rasi. This will be past (gata) phala. 
(172) 

For extra degrees, they are multiplied by 
difference for the khandaphala and divided by 60. 
Result is added to degrees obtained earlier. 
Remainder is multiplied by 2 and added to 
mandaphala khanda. This way, mandaphala of a 
graha is calculated, which is added or substracted 
according to rules earlier explained. 

Manda kendra gati multiplied by difference 
of khandaphala and divided by 225' (3 °45'), is 
nianda gati phala between two khandas. (173). * 

In appendix, parocca khandaphala of mangala, 
budha, sani also have been given. Khandaphala 
difference and ucca gati at end of khanda has also 
been written. From them parocca phala is calculated 
a nd is added or substracted from manda kendra 



4 



234 Siddhanta Darpana 

of mahgala, budha, sani or budha sighrocca, we 
get sphuta gati corrected for parocca. (174) 

Sighra khanda table also is prepared for 48 
parts (khanda of 180° i.e. 1 part of 3° 45'. Sighra 
kendra is found by substracting manda sphuta 
graha from sighrocca. Sighra kendra in 6 rasi's 
beginning with mesa is caled gata and in 6 rasis 
beginning with tula it is called gamya. Rasi, degrees 
etc. of kendra are multiplied by 8 and divided by 
60 to get the khanda number (because there are 8 
parts of 3° 45' each in 1 rasi of 30") as before. 
Khanda phala of completed parts is corrected for 
fraction parts by addition if khanda phala is 
increasing, or by substraction if it is decreasing. 
This is sighra phala. (176) 

If sighra kendra is in first 6 rasis/ khanda 
phala is added (to manda sphuta graha), or in 
other six rasis it is substracted. This way madhyama 
graha is made sphuta by sighra phala half, half 
mandaphala, full manda phala and full sighra 
phala. (177) 

For ravi, candra and marigala, manda paridhi 
is different for different quadrants. So their 
mandaphala also has been written for 48 parts of 
180° like sighra phala. For value between two 
khandas, we add fraction of khanda phala dif- 
ference if khanda phala is increasing. It is sub- 
stracted when khanda phala is decreasing. Manda 
phala is never retrograde. (178) 

For manda phala of mangala, there is no need 
of calculation between 22nd and 28th khandas. For 
that interval khanda phala is constant 11*2'47". 
(174) 



235 
True Planets 

Sighra kendra gati is substracted from khanda 
t>hala and result is divided by 225'. Half of the 
result is added to madhya gati, if sighra phala is 
increasing. It is substracted, if sighra phala is 
decreasing. We get 1st corrected gati. (180) 

First gati is multiplied by manda phala 
dfference between two khandas in which 2nd 
manda kendra lies and divided by 225'. Half of 
the result is added to first gati, if manda kendra 
is betwen 90° to 270°, otherwise it is substracted. 
We get second gati. (181) 

2nd gati is multiplied by manda phala 
difference for 2nd graha and divided by 225. Result 
is added or substracted from second gati to get 
third gati sighra. (182) 

3rd gati substracted from sighrocca gati gives 
fourth sighra kendra gati. This is multiplied by 
khanda phala difference of 3rd graha (manda 
sphuta) and divided by 225'. Result is added to 
3rd gati, if sighra kendra is in 90° to 270', otherwise 
substracted. We get spasta daily gati. If it is 
negative, graha is vakri (retrograde). (183) 

For mangala, budha and sani, vakra mandocca 
gati is added to 1st and 2nd gati and marg£ 
mandocca gati is substracted to find the kendragati 
from mandocca. Mandaphala of this manda kendra 
is found for second and 3rd gati. sighra.^ (184) 

If margi (forward) mandocca gati is more than 
first gati, then first gati is substracted. From 
remainder second gati will be calculated. Similar 
method is used for finding 3rd gati. Gatiphala is 
corrected in reverse manner i.e. substracted for 



236 Siddhanta Darpana 

manda kendra between 90° to 270° and added for 
other values. (185) 

This way we get second and third gati of the 
three planets mangala, budha and sani. (186) 

If 1st gati of budha and sani is vakra and less 
than mandocca gati, then it is substracted to get 
second gati. If vakra gati is more, mandocca gati 
is substracted from it but mandagati phala is added 
or substracted in opposite order. (187) 

Verse 188 - If in chart of khanda phala, some 
khanda phala is missing or unclear, then its khanda 
number is multiplied by 225' and for kendra of 
that kala, we find bhuja and bhuja koti. 

Verses 189-191 - Difference from sphuta surya 
in degrees is given at which a graha sets due to 
sun rays 

Vakri Sukra 7°, Sukra (margi) 9° 

Guru 10°, Chandra 11° 

Budha 12°, Sani 14°, Mangala 16° 

These values in degree are multiplied by 1800 
and divided by rising time of the rasi in which 
sayana sun is situated. This will be ksetramsa. If 
it is in west, then 6 rasi is added to the result. 
Then ksetransa is substracted from (sayana sun + 
6 rasi.) 

When mangala, guru and sani are less than 
ravi by at least the ksehansa, they rise in east 
before sunrise. When they are ahead of sun by 
ksetramsa, they set in west after sun. (Thus they 
are visible only in night). When vakri budha and 
sukra are behind ravi by this ksetransa, they set 
in east and when ahead of ravi, they rise in west. 



True Planets 237 

(Just before sun rise, since sun is coming upon 
horizon, they go down being vakrl. During night, 
they are visible when sufficiently away). Similarly, 
they rise in west just after sun set when vakri). 

Notes : (1) Rising times of rasis is explained 
in Triprasnadhikara. Briefly, rasis rise in different 
time because it is oblique with equator (23-1/2)°). 
At places farther from equator, obliquity rises and 
difference in rising time of rasis increases. This 
calculation is done for sayana surya, because surya 
goes on equator when sayana surya is at 0° or 
180°. Roughly the planets are assumed in same 
plane as sun, as their inclinations to ecliptic are 
very small. So rising time for their difference along 
ecliptic will be same as rising time of sayana sun 
for that rasi. Since rising time is given for 1 rasi 
of 1800 kala in asu, equivalent difference on ecliptic 
is given by multiplying given degrees (kalamsa) by 
1800 and divided by rising time of rasi. 

This is almost same as kalamsa, being its 
projection on ecliptic. 

(2) When planets are behind sun, they rise 
before sun in east, if difference is more than 
kalamsa. Being behind, earth horizon in east meets 
them after wards. Vakri budha and sukra have 
already been explained. 

Verses 192-193 : Finding time of udaya or asta 

From the kalamsa given we can calculate the 
time in days since when graha has set or risen 
(heliacally). If their difference with sun is more 
than Kalamsa, the planet has already risen or set. 
If it is less than kalamsa, the time to reach kalamsa 



238 Siddhanta Darpana 

can be calculated, which will be days after which 
planet will rise or set. 

(Difference of planet and sun - kalamsa) is 
divided by difference in speeds of sun and the 
planet. The no. of days will be found since when 
planet is rising (or setting) or after which it will 
rise again. 

Verse 194 : Start and end time of rising and 
setting of planets should be written in the practical 
calender, because it is very difficult to find it by 
drk karma. 

* 

Verse 195 : In appendix, khandaphala and 
their differences are given. Similarly differences of 
gati phala, and karna (in kala) also should be 
calculated and written. Sighrakarna, gati and 
sphuta positions etc will be found by values given 
for places just before the given position. Difference 
of phala is to be added or substracted when the 
value (phala) is increasing or decreasing. 

Verse 196 : Frequency for finding true 
positions - Sun and moon should be made sphuta 
every day at sunrise time. At end of a paksha, all 
graha should be made sphuta. Budha should be 
made sphuta in middle of paksa also (i.e. every 
week). When a planet becomes margi from vakri 
or vice versa, or changing from one rasi, naksatra 
to another, or start of rising time or setting should 
be calculated more accurately by method of 
successive approximations. 

Verse 197 - There are 200 kala (minutes of 
arc) in a quarter of a naksatra, 800 kala in a naksatra 
and 1800 kalas in a rasi. To find the days since 
when the graha is in a particular rasi, naksatra or 



True Planets 239 

quarter of a naksatra, we take the difference of 
rasi etc of graha and the rasi etc of the beginnig 
of rasi, naksatra or its quarter. The difference is 
divided by sphuta gati kala. Result will be days 
etc since when the graha had entered that rasi etc. 
When graha is less than rasi of naksatra etc, the 
reverse difference will be divided by sphuta gati. 
Result time in days etc. will give the period after 
which graha will enter that naksatra etc. When 
graha is vakri, opposite process will be done. 

Notes : (1) Ecliptic of 360° has been divided 
into 12 rasis and 27 naksatra of equal interval. 
Hence 

1 rasi = 30° = 1800' Kala 

1 naksatra = 13° 20' = 800' kala 

1 naksatra quarter (1/4 or pada) ■ 3° 20' = 200' 
kala 

(2) Rasi's starting from 0° of ecliptic are 

(1) mesa (2) vrsa (3) mithuna (4) karka (5) 
sirnha (6) ka'nya (7) tula (8) vrstika (9) dhanu (10) 
makara (11) kumbha and (12) mina 

Naksatras starting from 0° of ecliptic are 

(1) asvini (2) bharani (3) krttika (4) rohini (5) 
mrgasira (6) ardra (7) punarvasu (8) pusya (9) aslesa 
(10) magna (11) purva phalguni (12) uttara phalguni 
(13) hasta (14) citra (15) svat! (16) visakha (17) 
anuradha (18) jyestha (19) mula (20) purva asadha 
(21) uttara asadha (22) sravana (23) dhanistha (24) 
satabhis (25) purva bhadrapada (26) uttara 
bhadrapada (27) revati. 

(3) Within a rasi or naksatra a graha can be 
assumed to have the same true motion hence the 
formula uses the relation- 



240 Siddhdnta Darpana 



7*. 



1 



Distance in kala = days X speed per day in 
kala. 

(4) Candra moves faster and position of candra 
and sun are to be known accurately for start of 
day, tithi etc. Hence they are to be calculated each 
day. Other planets are not so important so they 
can be calculated each paksa (fortnight). Budha 
moves faster, hence its calculation should be done 
twice in a fortnight. 

(5) For change of vakri or margl gati or rising 
or setting times, the speeds change within a day 
also. Hence calculation needs to be made accurate 
by method of successive approximation. 

Verse 198 : Dainika spasta gati of a graha 
can be found roughly by taking difference of spasta 
graha at beginning and end of the paksa (fortnight) 
and dividing it by number of days in it (round 
figure of 14 or 15 when days are counted from 
sunrise to sunrise) Difference between spasta graha 
on two successive days at sunrise is more accurate 
dainika gati which is useful for calculation. Both 
differ very little, so very little error is made if we 
take average daily speed for a paksa. If the two 
are different, then method of successive approxima- 
tion is used. 

Verse 199 - Fourth slghra kendra is calculated 
at the end of every paksa. As already stated, slghra 
kendra of graha for which it becomes margi or 
vakri, its rising and setting has been given ir^ 
appendix. To find the position of slghra kendra at 
any time between paksa ends, divide the difference 
beetween values at end with days of paksa and 
add them proporitionately for the time passed. 



■1 

rJ. 






y 
V"- 



True Planets 241 

Veise 200 : 21,600 kala divided by 30, 27, 12, 
27 and 60 gives measures of tithi, naksatra, rasi, 
yoga and karana, i.e. 720, 800, 1800, 800 and 360 
kalas. 

Notes : (1) Tithi, naksatra, yoga, karana and 
vara are five parts of a calender - hence it is called 
pancanga. Vara is successive counting for days 
starting from sunrise, hence no calculation is 
needed. 

(2) Definitions - 'rasi' is 30° part of the ecliptic 
where planets move. Rasi of a planet means its 
completed rasis from 0° of ecliptic as well as 
degrees, minutes, seconds, lapsed in the current 
rasi. Though it is not part of pancanga, it is used 
to calculate all other parts. 

Naksatra is found by dividing ecliptic into 27 
equal divisions of 13°20' each (total 360° = 27 X 
13° 20') Each part is naksatra. 'Naksatra' mentioned 
in pancanga means the naksatra which is occupied 
by moon at a particular time. 

i.e. the time when moon goes one circle more than 
sun. It is measured usually from the time when 
sun are moon are together, i.e. difference between 
their rasi is 0°. That is start of first tithi called 
amavasya, i.e. when sun and moon live (vasa) 
together (ama = amity = closeness) Month can also 
b e counted from time when sun and moon are in 
opposition (i.e. 180° away) Then full moon is seen, 
so that is end of purnima tithi. The two systems 
°f lunar month are called amanta (ending with 
a navasya) or purnanta (ending with purnima). 



242 Siddhanta Darpana 

Tithis are not counted serially from one to 30 in 
lunar month. They are counted from 1 in each half 
(Sukla = bright and krsna = dark) In sukla paksa 
last tithi is written 15 and in krsna paksa it is 
written 30 (denotin end of month). 

Since 360* difference between moon and sun 
causes 30 tithis, 1 tithis is result of 12° difference. 
Thus difference of 0° to 12° is 1st tithi in sukla 
paksa after amavasya, 12° to 24°/ 2nd tithi etc. 
upto 180° the paksa will be sukla paksa with 15 
tithis. Between 180' to 360* difference it will be 
krsna paksa with 15 tithis. Thus the number of 
complted tithis 

Moon - sun 

" 12° 

Fraction will give the part elapsed in the 
current tithi which is next after completed tithi. 

When the quotient is more than 15, than 15 
is substracted to know tithi of krsna paksa. 

Karana is half part of tithi, caused by 6" 
difference between moon and sun. Thus completed 
karana since amavasya end 

Moon — sun 

These are not counted from 1 to 60 in a 
month, but there is rotation of 7 karanas like 7 
week days, 8 times in a month and 4 remaining 
karanas are given separate names fixed at both 
ends of a month. This is explained later in detail. 

Karana and tithi both indicate the phase of 
moon, i.e. the fraction of its disc which is 
illuminated. Naksatra and rasi of moon (or any 



True Planets 243 

other planet) can also be physically seen. But yoga 
is not a physical quantity. It is only a mathematical 
function given by sum of rasi etc of moon and 
sun (for tithi and karana, their difference had been 
taken). However, one full revolution of moon + 
sun is not divided into 30 parts like a tithi, but 
in 27 parts only like a naksatra. Thus for each 
increase in sum of moon and sun by 13 °20' one 
yoga passes. Thus number of completed yogas 
counted from time when sun of moon + sun was 
360° or a is 

Moon + sun 

13° 20' 

List of yoga is given later. 

(3) In a full circle there are 21,600 lipta or 
kala. Hence measure of naksatra etc is found by 
their total number in circle by which 21,600 is 
divided. 

Verse 201-202 - Calculation of tithi - 

Time lapsed (gata kala) and remaining time 
(gamya kala) of the current tithi is found by 
dividing difference of moon and sun in kala by 
720 kalas. Remainder is converted to vikala (on 
multiplication by 60). This will give gata kala. 
Dainika gati of ravi and candra is found by 
difference of current day and next day's position. 
Gata or gamya tithi is divided by difference «of 
dainika gati of moon and sun. This will give value 
m danda etc. (when gata tithi was in vikala). This 
is rough approximation, sufficient for normal work. 
^ this we have used dainika gati for 1 savana 
dl na in stead of gati in 1 tithi. If further accuracy 



244 Siddhdnta Darpana 

is needed, we find gati of a tithi from dainika gati 
and ravi, candra are further corrected. 

Verse 203 - Lapsed or remaining time in rasi 
or naksatra — Sphuta kendra is converted to kalas 
and divided by 800. Quotient will be number of 
past (gata) naksatras counted from asvini. By 
adding 1, we get the number of current naksatra. 
Remainder is the lapsed part (in kala) of the current 
naksatra. Substracting this from 800' we get 
remaining part. It is multiplied by 60 to make vikala 
and divided by dainika gati (in kala). This will give 
lapsed (or remaining) time of naksatra in danda 
etc. 

Sphuta candra converted to kala and divided 
by 1800 kala in a rasi gives number of completed 
rasis. By adding 1 to quotient we get the number 
of current rasi, counted from mesa. Remainder will 
be lapsed part (in kala) of the current rasi). It is 
substracted from 1800' to give remaining (gamya) 
part. Gata or gamya part is converted to vikala by 
multiplying with 60 arid dividing by spasta dainika 
gati of candra. We get gata or gamya kala of the 
current rasi in danda etc. 

Note - Gata or gamya part (in kala) — x = 60 x vikala. 

kala 
Dainika gati = Difference in position in 1 day = -r— 

gata part xkala 

Hence *j — = — x day = 60x danda 

gab Kala/ day J 

Hence n is converted to vikala before division by gati. 

Verse 204 — Calculation of yoga 

Add the rasi of sphuta candra and surya. If 
it is more then 12 rasi's, substract 12 rasi from the 
sum. It is converted to kala and divided by 800 = 



True Planets 245 

nC . of kala in a yoga. Quotient will be number of 
completed yoga counted from viskumbha. Add 1 
to it, we get number of current yoga. Remainder 
gives part of yoga lapsed in kala. By substracting 
it from 800', we get remaining part of current yoga. 
Gata or gamya kala is multiplied by 60 to make it 
vikala and divided by sum of dainika gati of sun 
and moon. We get gata (or gamya) time in danda 
etc. 

Note : List of yogas .— (1) viskumbha (2) priti 
(3) ayusmana (4) saubhagya (5) sobhana (6) 
atiganda (7) sukarma (8) dhrti (9) sula (10) ganda 
(11) vrddhi (12) dhruva (13) vyaghata (14) harsana 
(15) vajra (16) siddhi (17) vyatfpata (18) variyana 
(19) parigha (20) siva (21) siddha (22) sadhya (23) 
subha (24) sukla (25) brahma (26) aindra (27) 
vaidhrti 

Verse 205 - Calculation of karana 

■ 

Add 360 kala to spasta ravi in kala. Deduct 
the sum from sphuta candra in kala. Divide the 
difference 360 i.e. no. of kala in a karana. Quotient 
is divided by 7. Remainder is number of completed 
karana. By adding, we get the current karana. 
Karana starts from second half of sukla 1st day 
with 'Bava'. After end of seventh karana, again 
first karana 'bava' starts. In 30 tithis of candramasa, 
there are 60 karanas. 7 Karanas are repeated 8 
times. Remaining 4 karanas are fixed (sthira) wlfich 
are sakuni, naga, catuspada and kinstughna. 

Note : (1) Moving karanas start after 1st half 
°f 1st tithi (sukra 1st tithi) has already passed. 
Hence 360 kala is added to ravi so that in difference 
from moon, 1 karana is deducted. 



246 Siddhanta Darpana 

(2) Seven moving karanas (chala karana) are 
- (1) bava, (2) balava (3) kaustubha (4) taitila (5) 
gara (6) vanija (7) visti or bhadra. Last karana is 
considered inauspicious for good work. Similarly 
Sunday was not supposed a day for doing work 
out of seven week days. 

(2) Sthira karanas sakuni, naga, catuspada and 
kinstughna start from krsna 14th second half, 30th 
(15th krsna both halves) and sukla 1st tithi. 

(3) In vedanga jyotisa, 11 karana or half days 
were deducted from solar half year (equinox to 
next eqinnox in opposite direction) to make it equal 
to lunar month. 371 tithis in a solar year are 
divisible by 7, though 365 days are not divisible, 
hence fraction of weeks remain. Similarly in half 
year, karanas (half tithis equal to 371) are divisible 
by 7. Out of 11 karanas last 4 are fixed, as in a 
month also 4 remain after 7X8 cycles of 56 karanas. 

Verse 206 - If at time of sunrise, the total 
gata and gantya kala of tithi (720) is more than 
the difference in dainika gati of candra and ravi 
(i.e. difference is less then 720' per day), then tithi 
is long (tithi vrddhi) i.e. more than 60 dandas. 
Tithi vikala 720 x 60 divided by difference of 
candra and ravi gati, we get duration of tithi in 
danda etc. If it is more than 60 danda then there 
is tithi vrddhi, otherwise tithi ksaya occurs. 

Verse 207-209 - Extra and ommitted candra 
months- When in a candra masa, there is surya 
sankranti (i.e. surya goes from one rasi to another), 
then it is called suddha candra masa (i.e. normal 
month). When there is no surya sankranti (i.e. 
surya remains in same rasi), it is called extra month 






True Planets 247 

(mala or adhika masa.) Next amanta month is called 
normal candra masa. When there are two 
sankrantis of surya in a candra masa, it is called 
ksaya masa (lost month) - i.e. next, candra masa 
is not counted. Before and after ksaya masa, within 
4 months there are one mala masa each i.e. two 
mala masa in that year. First mala masa is called 
sansarpa, ksayamasa is called amhaspati and later 
malamasa is called mala. Both mala and ksaya masa 
are prohibited for any auspicious work. (207) 

In veda and smrti, the works which are 
prescribed, monthly and annual sraddha can be 
done in sansarpa or amhaspati, but not in the later 
malamasa. Malamasa is counted as a month for 
annual sraddha of a dead man, when it comes 
within start and completion of a month. New work 
is not started in a malamasa, but work started 
earlier can be continued. The following works can 
be done in a malamasa- 

Bath during eclipse, charity, observing rare 
yogas (auspicious times), sudden works, promised 
work, coronation, santi, pusti karma, functions 
related with child birth, sraddha etc. (208) 

A ksaya masa is repeated after 141, 122 or 19 
years. In current year (1869 when book was written) 
mandocca of sun was in mithuna, hence in 9 
months from phalguna, a mala masa is probable. 
3rd months after karttika may be ksaya masa, 
Magna month may be ksaya or adhika. 

Notes : (1) A lunar synodic month is 
approximately 29.5 days long, where as surya 
remains in a rasi of 30° for 30.4 days. Thus lunar 
month is completed earlier and after about 30 



24# Siddhdnta Darpana 

months extra days in solar month will amount one 
month and sun will not cross to next rasi. Example 
of mala masa is explained below - 

a p y d y 

Sravana Bhadra Asvina Karttika Margslrsa Pausa 
A B C D E F G 

ABCD are kranti of sun. Signs on upper 

part denote start of a lunar month. In Bhadra there 
is no sankranti so it is a mala masa. 

(2) Frequency of malamasa - There are 1593336 
malamasa is 51840000 solar months of a yuga i.e. 
66389 adhikamasa in 2160000 solar months. 

66389 __LJ__LJ_J_1 1 

2160000 " 32 ' +1 ' +1 ' +8 ' +1 ' +1 ' +5 

n L 1 1 2 13 15 25 

Convergents are -,-,-,— , — / ^ I 

1 J 2 
— and — are on either side of the true figure. 

Hence adding numerator and denominator 
both, we get a better approximation. Thus 3/98 is 
ratio of adhika masa i.e. 3 adhika masa in 98 
months (solar). 

(3) Adhika masa and year — There are 
1,593,300,00 adhika masa in a kalpa of 4,320,000,000 
years 

i.e. 5311 adhika masa in 14400 years 

14400 1 1 1 1 1 11 i i 



5311 1+ 2+ 2+ 6+ 1+ 1+ 7+ 8+ 2+ 

Successive Convergents are 



True Planets 249 

2 1 3 19 122 141 
T ' 3 ' 8 ' 7 ' 55 ' 62 

Thus there are approximately 7 adhika masa 
in 19 solar years which was used in vedanga jyotisa 
(Rk veda). This was known in Romaka siddhanta 
and was called Metonic cycle in Greece. 

Next approximations also indicate possibility 
of ksaya masa in 19, 122, 141 years. 

Verse 210 - Thus the rough pancanga with 
its components like tithi and naksatra is completed 
which may be accepted by the learned and they 
may perform every year the daily, occasional and 
conditional functions, fasting days, sraddha, fes- 
tivals etc. according to this pancanga. This may do 
good of world as it is according to jyotisa samhita 
and well thought of. 

Verses 211-212 - For daily auspicious functions 
I am preparing this pancanga with positions of sun 
and other planets. While doing the work I pray 
to lord Jagannatha who is on nilacala shining like 
black soot (for eyes). 

Thus the fifth chapter describing true planets 
with their khanda phalas is over in siddhanta 
darpana written for education of children and 
calculation as per observation by Sri Candrasekhara 
born in a famous royal family of Orissa, 




Chapter - 6 

CORRECTIONS TO MOON 

Scope - Accurate panjika and further correc- 
tion to motion of Moon 

General Introduction 

(1) Equation for elliptical orbit round earth. 

Eccentricity of moon is 0.0548442 = e 
So e 2 = 00.0030079, e 3 = .00016496 
e* = .00000905 

Higher powers e 5 etc are very small and can 
be neglected. Thus 6 measured from mandanica or 
perigee is given in terms of position m of mean 
planet as 

* = m + (2e-±e 3 + ^) sinm 



96 

5 2 11 4 „ e 6 



+ i 7 "24* + 17 I^)sin2m 

13 3 43 5 
{ 12 6 ~ 64* ^ Sin 3m 

103 4 451 6 1097 e 5 

( 16* 480* > Sm4m + ~lio- sin 5m ... 

= m + (0.1096884 - 0.00004124) Sin m 
+ (0.0037599 — 0.00000415) sin 2 m 
+ 0.0001787 sin 3 m + 0.0000097 sin 4 m 
= m + 0.10964716 sin m + 0.00375575 sin2 m 
+ 0.0001787 sin 3 m + 0.0000097 sin 4 m 



Corrections to Moon 251 

The sine ratios in radians are converted to 

1 i_ 180 ° 

kala fer degree) by multiplying with ~ ir x 60 = 

3437.75 kala or 206265 vikala. Then 

$ = m + 376'56". 4 sin rn + 12'54". 7 sin 2m 
+ 36". 9 sin 3 m + 2".0 sin 4 m 

Here m has been calculated from nica or 
prigee. If it is calculated from apogee or mandocca, 
then 

6 = m - 376'56." 4 sin m + 12'54"7 sin 2 m 

- 36".9 sin 3 m + 2". sin 4 m 
Here m on right side is manda kendra - i.e. 
distance of madhya graha from mandocca of moon. 
Remaining terms are mandaphala. 

When 6 = 90°, sin m = 1 and sin 2 m = 
Then highest mandaphala depends only on 
its first term 377' approximately or 6*17'. But our 
astronomers have taken highest mandaphala about 
5° only (radius of mandaparidhi of 32°). However, 
on new moon or full moon day, when moon is 
90° away from mandocca, then it is V2(Y ahead 
of its calculated position. When moon is 270" ahead 
of mandocca or 90* from nica then it is 1° 20* behind 
its calculated position. Thus in both situations 
mandaphala correction is 6'16'56".4-r20' = 4°56'.4 
(correction is-ve for m = 90° and positive f or m = 
270°). Thus maximum mandaphala is about*5° only 
as observed. 

However, in middle of a paksa i.e. on 8th 
day, if this mandaphala correction for manda 
kendra 90° is taken as 5°, then observed moon is 
3° behind calculated moon or 8* behind mean 



252 Siddhanta Darpana 

moon. Thus cauculations in our siddhanta were 
true for purnima or amavasya when eclipse is to 
be calculated. One reason for such neglect is that 
accuracy is needed only for eclipse, other reason 
is that observations were done ony on purnima or 
amavasya days or more accurately at time of eclipse. 
This is still followed by muslims and even now 
eclipses are studied for more accurate observation. 

(2) Deviations in moon position due to effect 

of sun - Effect of sun is three types 

(a) Attraction component of sun on moon in 
direction of earth moon radius, elongates the orbit 
in the direction of sun and away from it. It changes 
eccentricity of orbit and is called evection term. 
Since it changes eccentricity of orbit, called 'cyyti' 
it was called 'cyuti' sanskara by Sri Venkatesa 
Bapuji Ketakara in his Jyotirganita. Since it changes 
angle from mandocca (or Tunga = top), it has been 
called Tungantara' sanskara in siddhanta darpana. 

(b) Component of sun's attraction on moon 
in direction of moon's motion advances it towards 
sun, which is maximum in middle of a paksa and 
nil at its ends. This varied speed, hence it was 
called variation. Its frequency is in 1 paksa, hence 
it is called paksika sanskara in siddhanta darpana. 
Sri Ketakara called it tithi sanskara because it 
depends on tithi of the paksa. 

(c) Due to difference of sun's distance from 
earth or moon depnding on its direction from earth, 
its attraction force on moon varies in a period of 
1 year. This is called digamsa sanskara as it 
amounts to 1/10 of sun's equation. This is also 
called varsika sanskara because its period of 
variation is one year. 



i 
■n 



Corrections to Moon 



253 



Figure (1) (a), shows force of attraction G due 



s* 




Fig 1 (a) Fig 1 (b) Fig 1 (c) 

Figure 1 - Effect of sun's attraction on moon's orbit 



to sun. In positions A and B which are near to 
sun compared to earth, extra attraction on moon 
is in direction of sun. In position C and D of 
moon, away from sun, the difference in force 
compared to earth is away from sun. Force of 
attraction G has two components, its component 
R is reducing the pull of earth on moon acting in 
opposite direction. Thus distance of moon increases 
from earth. This increase is maximum for positions 
Mi and M3 and nil for positions M2 and M4- Thus 
in Fig 1 (b), when major axis is in direction of 
sun, the axis will become longer and its eccentricity 
will increase. 

In fig 1 (c), the distance perpendicular to 
ma Jor axis in sun's direction will increease, due to 
which moon orbit will become round. Then 
e ccentricity will decrease. Thus correction in man- 



254 

Siddhanta Darpana 

daphak due to eccentricity will increase for fig (1) 
(b) and decrease in position of figure (c). 

and ^T V ° ne ?u T iS maximum for position M« 
and mcreases the speed in middle of krsna paksT 

of Z C T *?* beC ° mes zero at M > « *e force 
of attraction ,s totally in direction of EM,, aiS 

other component is zero. It increases in value from 

m! o M T 3gain d6dineS to 2er <? at M» ^om 
M, to M 2 it is against the direction of motion. 

It is in direction of motion between M,, M, 
(decreasing) and again increasing upto M< b* 
against the motion. 

A««5? Correction by different authorities - 

of m S ™ ^° dem astronom y, Principal terms 
or moon s motion are - 

= m 

■+ (377'19".06 sin m + 12'57".ll sin 2 m 

correct/ 9 Sln 3 m + 2 "° ^ *»> -ndaphala 

+ 1-16-26" sin [(2 (M-S) - m] - - Evection or 

Tungantara 
+ 39'30" sin 2 (M-S) - - variation or paksika 

+ ll'lO" sin (manda kendra of sun) - - - 
Annual or digamsa 

onlv 'Si Carly / S u°f ° merS ° f ^ cognised 
™LK, mand ^ ya correction, (equation of 
centre), but instead of its value to be 377' sin m 
they took its value 301' sin m, by includtag effeS 

301' sin m + 76' [(sin m + sin {(2 (M-S) - m ]]+ 



Corrections to Moon 255 

= 301' sin m + 152' sin (M-S). Cos (S-a) 

Here M = mean Moon, S = True sun, a = 
moon's perigee from which angle m has been 
measured. Thus m - (M-S) = S - (M - m) =s-a 
in cos term above. 

Value of 1st correction to moon was the 
following according to different authors 

Aryabhatiya 300'15" Sin m 

Khanda khadyaka 296' sin m 

Uttara khanda khadyaka 301'.7 sin m 

Brahma sphuta siddhanta 293'31" sin m 

Greek value 300'15' sin m 

Siddhanta darpanaNj00'49".5 sin m 

Surya siddhanta 302'23".66 sin m 

Bhaskara H, 301'46".8 sin m 

(a) Second correction term by Manjula (932 

AD) 

In Laghumanasa, 1st mandaphala correction 

of moon has been given as 

488' sinm 
— degrees 

97' + — cos m 

where m is mandakendra measured from 
apogee. Thus maximum value of mandaphala is 
for m = 90°, 

488 

^ degrees = 301'50" 

Second correction has been given by 

8° 8' cos (S-U) (True moon - 11) x 8° 8' sin 
(M-S) 



Siddhdnta Darpana 

where S,M,U are true sun, true moon and 
mandocca of moon. 

For simplicity, daily motion of 790 °35" of 
moon is taken as true motion, then this becomes 
8 B 8' x 8-8' x V 11' cos (S-U) sin (M-S) 

= 144-26' cos (S-U) sin (M-S) - converted to 
minutes, 2nd correction (l) 

Thus Manjula's correction is sum of two 
correction - 

(i) 76' sin (M-U) - part of the mandaphala 
(ii) 144-26" cos (S-U) sin (M-S) - - - evection 

term which was not mentioned by previous 

astronomers. 

Plotemy had given maximum value of 2nd 
correction as 159' but didn't give any formula (150 

Astronomer Yallaya gives credit of this dis- 
covery of these corrections to Vatesvara (904 AD) 
but this has not been found in Vatesvara siddhanta 
the available book. ' 

This appears in exactly the same form in 
karana - kamala-martanda of Dasabala (1058) 
Subsequently it occurs^ in equivalent forms in 
siddhanta sekhara of Sripati (1039 A.D), Tantra 
Sangraha of Nila Kantha (1500 A.D.), uparaga kriya 
krama of Narayana (1563 A.D.) Karanottama of 
Acyuta (1621 AD) and lastiy in siddhanta darpana 
of Candrasekhara (1869). " 

Equation (1) of Manjula is correct but constant 
is 8, less. 

Snpati's second correction amounts to the 
following correction term. 



Corrections to Moon 257 

160' cos m sin (mandaphala) x 
1 - cos (mandaphala) 

Mandakarna - R 

where R = radius 3438' 

This is same as Manjula's equation except that 
the constant is now. 8' more, instead of 8' less 
earlier. 

(b) Bhaskara II - Bhaskara II wrote a separate 
work called 'Bijopanaya' about corrections needed 
in true planets. Stanza 8 of the work starts with 
statement - 

I have seen maximum difference between 
calculated and observed positions to be ± 112' 

When moon is one quadrant ahead of man- 
docca and sun is half aquadrant ahead of moon, 
observed moon is 112' behind calculated moon i.e. 
negative error. 

When moon is 3 quadrants ahead of apogee 
and sun at half a quadrant behind her, the 
maximum positive discrpancy of + 112' is seen 

When eclipses of sun and moon take place 
and moon is at apogee and perigee, there is no 
error or bija. 

When eclipses take place at end of odd 
quadrants from apogee, error is negative equal to 

TA r, . and sun is 

When moon is at the apogee, and :.-;_ la 

ahead or behind by half a quadrant, descrepancy 

Same discrepancy is seen, when moon is at 
P e ngee and sun half quarter ahead or behind 

His first equation of mandaphala was correct 



258 Siddhdnta Uarpana 

= - 301'46" sin m. 
. But after Bijopanaya he gave the equation 
- 379'46".8 sin m + 34'sin2 (M-S) 

where m is manda kendra, M and S are true 
moon and sun. His new equation totally missed 
the evection term, and it became more incorrect at 
eclipses; though his observations about error were 
correct. 

(c) Correction by Candrasekhara - 

His first equation of apsis (mandaphala) is 

(31° 30' - 30" cos m) 3438 sin m 

360° 
= - 300 / 49 ,, .5 sin m + 4'46".5 sin m cos m 
= - 300 / 49".5 sin m + 2'23".25 sin 2 m 

Though he has attempted to correct the 
second order of small quantities, his constant is 
too small (l/5th of the correct value). 

(2) Tungantara correction is of the form 

1W x 3438 sin (a - S - 90) 3438 sin (M - S) 

3438 X 3438 

(where a is apogee of moon) 

Moon's true deaily motion 

x ' ' 

Daily mean motion 

= - 16C cos (Sot) sin (D-0) 

Moon's apparent deaily motion 

Daily mean motion 

(3) Paksika equation or variation in Daily mean 

. . 3438' sin 2 (M-S) _,„_„ . -/wcv 
motion is ~> — — - = 38'12" sm 2 (M-S) 

90 



,3 



Corrections to Moon 259 

Here the constant is less by 1'18" from modern 
value. 

(4) Digansa sanskara for annual variation is 

1 12 x 3438 . „ 
■*■ — x zm sin S m 



~ 10 360 

(Sm = manda kendra of sun) 

= ± 11'27" sin Sm 

Modern value of the constant is ll'lO". Tycho 
found it to be 4'30". Horrocks' (1639)) found it 
11'51". He has indicated in the text that new 
equations were to correct the discrepancies ob- 
served by Bhaskara II, in which he was brilliantly 
successful. 

(4) Modern charts for calculating moon's 
position - 

Constants of moon's motion at 1900 AD, 0.0 
day epoch is 

Mean longitude L = 294°. 56984 + (1336 r) 
307.8905722 T + 0.00918333 T 2 + 0.00000188 T 3 

Mean anomaly M = 229°. 97832 + (1325 r) 
198°51'23".5T + 44".31T 2 + 0".0518T 3 

Mean longitude of node V = 259 Q 12'35".ll - 
6962911". 23 T + 7". 48T 2 + 0.008T 3 

For perturbations the constants are given by 
Hansen as- 

Ao = 69.80458 + (1148r) 55.37787761T + 
0\00881085T 2 + 0\0000011374958T 3 

Bo = 352.81434 + (2473r) 254" 23441630T 

+ °. 000420645 T 2 + 0* .00000301393 T 3 

Co = 204°. 85020 + (99r) 359*.051667T + 
0.0001988055T 3 



260 Siddhdnta Darpana 1 

D = 190 '.45443 + (1048r) 56 '.32271091 T + I 
0°. 007903044 T 2 + 0°. 00001 1374958T 3 

E = 354°. 45312 + (2373r) 255°. 17924960 T 
+ 0° 004405255 T 2 + 0°. 00000301393 T 3 
F = 341°. 85083 + (1131r) 172°. 20183595 T 
+ 0°. 00430092 T 2 + .000003347264 T 3 
Components of perturbation effect are - 
A = 4467" Sin Ao = 1.24083° Sin Ao 
B = 0.59583 Sin Bo 
C = 658" Sin Co = 0.18277 Sin Co 
D = 0.55 Sin Do 

Total effect of perturbation = G = A+B+C+D+E 
Perturbation in latitude is 
F = 0.1453 sin Fo 

From the value of these constants equation of 
centre and latitude is calculated. 

(5) Indian Charts - 

In India also many charts were prepared from 
time to time. Makaranda sararu was most famous. 
Candrasekhara has referred to tables of Kochanna 
of Andhra pradesh. Then in south India, specially 
in Kerala, vakya karana are very famous. Original 
Vakya karana was written for moon—called 
candravakyani by Vararuci, reputed to be in time 
of king Vikramaditya at start of Vikrama eera. Then 
Vakya karana was prepared in 13th century. Its 
writer is not known, but Sundararaja commentary 
is available. These books calculated the days from 
kaliyuga beginning. The moiton was calculated fof 
a convenient lump of days. For remaining number 
of days, the true position was calculated at about 



Corrections to Moon 261 

200-300 positions. These were indicated by (vakya' 
for each of position to be read in Katapayadi 
notation. This method could give correct position 
upto minute for 24 hour intervals. Madhava of 
sangamagrama in 1350 AD, prepared 'Sphuta 
candrapti' to calculate true moon upto seconds of 
arc at 9 periods in a day. His method was to 
calculate position of moon at equal intervals of 24 
hours from its mandocca position. Moon reaches 
from mandocca to mandocca in about 248 days, so 
248 vakyas are used. 

(6) Making of a calender - 

One of the main aims of astronomy is to find 
suitable measurement of time. A time scale to 
indicate past time since an epoch is a calender. 

Intervals of time which can be measured is 
one type of kala and its measurement is called 
'kalana' Thus 'calculate' means to count or to 
measure. In Arab, they were called 'kalamma' Work 
of 'kalana' is called chronology or calender. 

The flux of time is apparently without 
beginning or end, but it is cut up periodically by 
several natural phenomena- 

(i) by ever recurring alteration of day light 
and night 

(ii) by the recurrance of moon's phases 

(iii) by the recurrance of seasons 

These have been used to define natural 
divisions of time- 
Day - time of alteration of day and night 

Month - Complete cycle of moon's changes 

°f phase - 



262 Siddhdnta Darpana 

New moon to new moon (amanta month) or 

full moon to full moon (purnimanta) months. 

Year - Coming back of a season again and its 
smaller subdivision season. 

Standards for day - Day for purpose of regular 
works was counted from sun rise to sunrise in 
India and from sunset to sunset in west asia 
(Babylonians and Jews). West Asia was called 
'Asura' area and hence they were called nisacara 
(moving in night) because their day started from 
night time. Sunrise and sunset are convenient to 
see and day light only gives opportunity for doing 
works. 

Sunrise time varies according to position of 
sun in south or north hemisphere of sun. Variation 
of day length is more in places away from equator, 
being nil at equator. Hence for calculation purposes 
day was counted from midnight to midnight. 

Even midnight to midnight day varies, be- 
cause during this time earth makes one rotation 
arounds its axis with respect to stars and has to 
move further to catch up with movement among 
stars. This second component varies with distance 
of sun which varies in an elliptical orbit. Thus 
revolution of earth with respect to stars is taken 
as a better standard called sidereal day. An average 
of solar day (midnight to midnight) is used and 
called mean solar day. 

1 1 , . , 

365 - mean solar days = 366 - sidereal days 

1 
1 hour = — of mean solar day. 



i 



■2a 



■§ 



263 
Corrections to Moon 

Rotaton of earth = 23 h 56 m 4.100s mean 
solar time 

Sidereal day = 23h 56m 4.091s mean solar 

time 

Mean solar day = 24 h 3m 56.555s sidereal 

time Slight variation in rotation period of ear* 
and sidereal day is due to obliquity of earth, 
rotation being counted in the ecliptic plane. Even 
earth's rotation period is not constant but fluctuates 
regularly and irregularly by amounts of the order 
of 10* seconds. Regular slowing down of rotation 
period is 14 seconds per century due to tidal friction 
caused by difference of attraction force on sea water 
in different parts of earth. It is mainly by moon 
and l/4th by sun. Irregular variation is due to- force 
exerted by wind movements or unequal rate of 
atmospheric rotation and sea currents, both of 
which are caused by heat of sun. 

Month - 

Period from new moon to new moon varies 
from 29.246 to 29.817 days due to eccentricity of 
moon's orbit and other causes like effect of sun. 
Period of mean lunation is given by 
29.5305882-0.0000002 T days 
where T = no of centuries after 1900 AD. 
It may be noted that this is not the period 
of rotation of moon round earth. This is extra one 
round ahead of sun. When moon and sun are 
together, it is anuivasya (living together). Moon 
with its faster motion goes ahead in about 15 days 
by 180' when it is purnima (or full moon). After 
29.5 days it is again with sun. This rotation is with 



264 Siddhanta Darpana 

speed (moon-sun) and slower than moon's rotation 
in 27.3 days only. 

Year and seasons - 

1 year is one rotation of sun with respect to 
stars - it is called sidereal year. Seasons change 
according to position of sun with respect to earth 
in north south direction. It is perpendicular to 
equator twice in one year, while coming from south 
to north it is called vernal equinox and in opposite 
direction it is autumnal equinox. Equinox means 
equal day and night (nakta in sanskrta = night) If 
axis of earth is fixed, tropical and solar years will 
be same. But it rotates in reverse direction in a 
conical manner, thus equinox points rotates west 
ward making a rotation in about 25000 years. Due 
to this precession of equinoxes occurs. 

Tropical year = Sidereal year - speed of 
precession per year (crossing time by sun) 

Present values are 

Tropical year == 365.24219879 - 0.614 (t- 
1900)xl0" 7 days, where t = Gregorian year 

Thus it is 365.2421955 days = 365d 5h 48 m 
45.7 sec. 

Sideral year is 365.256362 days. 

Only tropical year corresponds to the seasons 

In addition to two equinoxes, we can take the 
points of longest day (in north hemisphere) where 
sun is northern most from equator i.e. summer 
solstice or the southern most position called winter 
solstice. 

As the day is counted from midnight i.e. 
lowest position of sun in east west circle, year can 



Corrections to Moon 265 

be counted from southern most winter solstice 
(which is lowest for northern hemesphere). This is 
like a grand day hence one tropical year is called 
a 'divya dina' (divine day). Since the grand day 
starts with winter solstice from vedic days, the first 
day 'christmas' is called 'bada (grand) dina'. 
Actually it is start of grand day. That month called 
margasirsa has longest nights hence it is called 
Krsna masa (or black month). Thus Krsna has 
compared himself with margstrsa month in gita. 
This has become 'Christmas' (Krsna masa). 15 days 
before start of margasirsa masa will be beginning 
of great usa (Twilight before sun rise), hence it is 
called 'bada osa' in local languages (like in Orissa) 

Problems in calender making - 

Civil calender for use in human life has 
following difficulties 

(a) Civil year and the month must have an 
integral numbers of days - perferably equal 

(b) Starting day of the year, and of the month 
should be suitably defined. The dates must 
correspond to seasons. 

(c) For the purpose of continuous dating, an 
era should be used and it should be properly 
defined. 

(d) The civil day, as distinguished from the 
astronomical day, should be defined for use in the 
calender. 

(e) If the lunar months have to be kept, there 
should be convenient devices for luni solar adjust- 
ments. 



266 Siddhdnta Darpana 

All the problems have not been solved till 
today. The errors in calculations also had to be 
corrected. Hence new calenders were started in 
different parts of the world by the intervention of 
dictators like Julius Caesar, Pope Gregory Xm or 
a founder of religion like Mohammad, or by 
monarchs like Melik Shah the Seljik or Akber. 

Owing to historical order of development, 
calenders have been used for double purpose. 

(i) of the adjustment of the civic and ad- 
ministrtive life of the nation. 

(ii) of the regulation of the socio religious life 
of the people. 

Divisions of day : 

Present division of day is in 24 hours. Minute 
divisions of 60 each called minutes and second 
division again by 60 called seconds. Thus 1 mean 
solar day = 60 x 60 = 86,400 seconds. Division of 
time and angle measures by 60 was because of 30 
days in a month and 12 lunar months in a year 
whose lowest common multiple is 60. A day has 
365 but approximate multiple of 60 is 360. Hence 
a civil year was taken of 360 days and a circle was 
divided into 360°. Thus sun will move about 1° 
in 1 day. In India, day was divided into divisions 
of 60 at each step as degree is divided. Thus 1' 
movement is in 1 day, V movement in 1 danda, 
1" movement in 1 pala and so on. 

Time was measured by length and direction 
of shadow of a pillar called gnomon. For equal 
time intervals, specially during night time, water 
clock etc were used. Improvements were done 
through pendulum clocks by Galileo, spring clocks 



267 

Corrections to Moon 

using balance wheel. Most occurate are quartz 
docks for normal use and ammonia clocks for 
scientific use. 

For practical watches of duty or shifts of work, 
a day was divided into 6 parts (3 parts m day time 
and 3 in night). After each Nerval a bell was 
rung In India there were 8 shifts in a day, hence 
the shift of 3 hours is called a 'prahara' i.e. when 
a bell is hit (prahara). A watchman remains on 
continuous duty for a prahara, hence he is called 
prahari. 

— day = 1 ghati is called so because water 

clock measured the time of its filling. Since it was 
shaped like a pitcher it is called 'ghati' (i.e. water 
pot). Hence watches are called 'ghadi' in India. 
When water clock in turned a second time it is 2 
ghati = 1 muhurtta (repeated turning of water 

clock). 

Watches observed in churches were 

(1) Martins - last watch of night. Monk got 
up 2 hours before sunrise 

(2) Prima - at sunrise 

(3) Tetra - Half way between sunrise and noon 
- time of saying mass. 

(4) Sext - at noon (hence the word siesta 
= midday rest) 

(5) Nona - Mid afternoon - Hence the word 
noon. 

(6) Vespers - An hour before sunset 

(7) Compline - at sunset 



26# Siddhanta Darpana 

In India mid day is 2 praharas after sunrise 
(i.e. 6 hours after), hence it is still called 'two 
pahars'. 

Day was divided into 12 parts in Babylone of 
30 gesh (4 minutes each). In each part approximate- 
ly 1 sign of zodiac will rise, it is like 12 divisions 
of year. In India rasi was divided into two parts 
(like day-night divisoin of day) called 'Hora' (short 
of 'ahoratra' i.e. day and night) Thus there are 24 
horas is a day night or 12 in day and 12 in night. 
This 'hora' has become hour. This was also used 
in Egypt and continues till today. 

Counting of days in a month : 

The ancient Iranian calender gave 30 names 
for each of the days of a month. It was not very 
popular as the list is long and difficult to remember. 
Hence a week of seven days was popular through 
out the world. Origin of week days has been 
explained by Varahamihira. Each hora (24 in a day 
is ruled by a planet. Planets are arranged in order 
of decreasing orbit or increasing speeds of rotation 
- Sard, guru, Mangala, surya (or earth), sukra, 
budha and Candra. In first hora of the day, lord 
of the day will rule. For example, Sani will rule 
1st hora on sani vara. On next day ruler will be 
25th planet in the order given above. Deducting 3 
cycles of 7 planets, 4th planet surya will be ruler 
of next day i.e. 1st hora on that day. So it is called 
ravivara or Sunday. Next day will be 4th from 
surya i.e. candra or moon called somavara or 
monday. 

Rulers of days are fixed for astrological 
purpose, hence it has astrological origin in India 



Corrections to Moon 



269 



and west. Ancient Egyptions had a ten days week 
(period in which sun covers 10° or l/3rd of a rasi 
called Dreskana in astrology) Babylonians started 
a month with new moon and marked the 1st, 8th, 
15th and 22nd days of the lunar month for religious 
festivals. This was a sort of weak of 7 days with 
one holiday. In Iranian calender in which 30 days 
had different names 8th, 15th and 23rd were called 
Diniparvana for religious practices. But last week 
in this system was of 9 or 10 days. In veda, sadaha 
has been mentioned, but this doesn't seem to 
indicate a six days week. It seems to be six extra 
days after 360 in a leap year called 'Gavam Ayana' 
every four years. The Jews reckon the days from 
Saturday and indicate them by numbers i.e. 1st, 

2nd 7th day. 

Seven days week was introduced to christian 
world by edict of Roman emperor Constantine in 
323 AD, who changed the Sabath day (Saturday 
for Jewish) to the Lord's day, Sunday. In India il 
has been first mentioned in Atharva Jyotisa and 
by Aryabhata. English names of week day hav€ 
originated from Teutonic deities which are countei 
parts of Roman planetary deities. 



Indian 


Childean 


Teutonic 


Roman 


names 


names 


dailies 


dailies 


Ravi 


Shamesh 


Sun 


Sun 


Soma 


Sin 


Moon 


Moon 


Mangala 


Nergal 


Tiu 


Mars 


Budha 


Nabu 


Woden 


Mercui 


Guru 


Marduk 


Thor 


Jupiter 


Sukra 


Ishtan 


Freya 


Venus 


Sard 


Ninib 


Saturn 


Saturn 



270 Siddhdnta Darpana 

It is note worthy that functions attributed to 
planets by Chaldeans are same as in Indian 
Astrology. 

Ahargana or heap of days - 

Count of days is used all over the world from 
a standard epoch to calculate the mean posotion 
of any planet. 

Mean position at required time 

= Mean position at initial epoch + daily motion 
x ahargana 

To make a uniform standard, a French scholar, 
Joseph Scaliger introduced in 1582, a system known 
as 'Julian days' after his father Julius Scaliger. The 
Julian Period is 

7980 years = 19 X 28 X 15 

19 is length in years of the Metonic cycle 

15 is length in years of the cycle of indication 

28 is length in years of the solar cycle 

It was found by calculation that, these three 
cycles started together on Jan 1, 4713 B.C. Julian 
period and days are counted from that day and 
the day is completed at noon time. This is the 
standard for astronomical calculations now. 

Julian days for some important epochs is given 
below 

Date Julian day 

Kaliyuga 17-2-3102 BC 5,88,465 

Nabonassar 26-2-747 BC 14,48,638 
Fhflippi 12-11-324 B.C 16,03,398 

Saka Era 15-3-78 AD 17,49,621 

Diocletian 29-8-284 AD 18,25,030 



Corrections to Moon 7 7 * 

Hejira 16-7-622 AD 19,48,440 

Jezdegerd 

(Persian) 16-6-632 AD 19,52,063 

Burmese era 21-3-638 AD 19,54,167 

Newar Era 20-10-879 AD 20,42,405 

Jalali Era (Iran) 15-3-1079 AD 21,15,236 

In India, siddhanta jyotisa uses ahargana from 
creation after wfeich 6 manus of 71 yuga each have 
passed, in current 7th manu 27 yuga have passed. 
In 28th yuga, Satyuga, Treta and divapara have 
passed. Present kali yuga started on 17-2-3102 B.C. 
Ujjain midnight. In this kali yuga is 4,32,000 years. 
Dvapara, Treta, Satya yuga are 2, 3, 4 times. A 
yuga is 10 times kali = 43,20,000 years. Before each 
manu there is a sandhya of a satyayuga. Thus 
years from creation till beginning of kali yuga are 
1, 97, 29, 44, 000 years. To find the ahargana for 
calculation, we deduct the years spent in creation 
= 47,400 divya years x 360 solar years. After this 
period all planets started from zero position which 
is called epoch. Ahargana at beginning of kaliyuga 
is 

714, 402, 296, 628 

Tantra granthas count the ahargana from kali 
era. Each karana book has used its own epoch. In 
present calculations Jan 1,1900 is important epoch. 
For this day Julian days are 2,415,021 and kali 
ahargana are 1, 826,556. 

(7) Solar calenders in History - 

(a) Egyptian calender - This has 12 months 
of 30 days each, starting from Thoth on 29th 
August as per Julian calender. This was old 
religious calender, hence extra 5 days were attached 



272 Siddhanta Darpana 

in the end which not part of any month. Since 
the year was short by 1/4 days from 365-1/4 days,] 
the heliacal rising of Sirius star would re-appear i 
at the beginning of year after 1460 years. This was^ 
called Sothic cycle as Sothis (Isis) was the goddess 
of sirius. In 22 B.C. the year started on 29th August 
the Pharoahs (kings) of Egypt tried to introduce 
leap year, but this never became popular. Ptolemy 
in 238 introduced a leap year, but old calender 
also continued side by side. Egyptians did not use 
any continuous era, but counted the number of 
years of each reign separately. For astronomical 
purposes, Nabonassar Era was used in Babylone. 
This was used as a reference by all countries for 
sumplicity 

(b) The Iranian calendar - Around 520 B.C. 
Darius introduced a solar calender like Egyptian 
with 365 days each. It had 12 months of 30 days 
each and each day had a specific name. The names 
are similar to vedic names. 5 days extra were 
attached in the end. Adjustment of 1/4 extra day 
each year was done by adding a month of 30 days 
in a cycle of 120 years. 

From 16-3-1079 A.D, Seljuc sultan Jelaluddin 
Malik Shah introduced a new calender Tarikh-e* 
Jalali, starting from 10th Ramadan of Hejira 471. 
It was 365 days year with 8 .intercalary days in 33 
years. The year started from vernal equinox day 
or next day. Its lenght was 365.242 42 days. 

Riza Shah Pahlavi introduced a strictly solar 
year and restored the old Persian names of month; 
in use before Darius. The year started from 21 or 
22 march. First 6 months were of 31 days each. 



Corrections to Moon 273 

Last month was of 29 days or 30 days in a leap 
year. 

Roman calendar (Christian Calendar) - The so 
called Christian calendar had nothing to do with 
Christianity. It was originally the calender of semi 
savage tribes of Northern Europe, who started their 
year some time before the beginning of spring 
(March 1 to 25) and had only 10 months of 304 
days, ending about the time of winter solstice 
(December 25). The remaining 61 days formed a 
period of hybernation when no work could be done 
due to on set of winter, and were not counted at 
all. 

This calender was adopted by city state of 
Rome and some modifications were made. Second 
Roman king of legendary period Numa Pompilius 
added two months (51 days) to the year in about 
673 B.C. making a total of 355 days. January 
(named after god Janus who faced both ways) and 
February were added in beginning and March 
became the 3rd month now. Number of days 
became now 29,28,31,29,31,29,31,29,29,31,29,29. 
Adjustment of the year to the proper season was 
done by intercalation of a thirteen month of 22 or 
23 days (caDed Mercedonius) after two or three 
years between February and March, the extra 
month was actually 27 or 28 days but, the last 5 
days of February due to be repeated after extra 
month, were not repeated. The correction at 
alternate year could have given 45 (22+23) days in 
4 years pr 11-1/4 days on average. Thus it made 
a year only one day longer than 365-1/4 days. But 
this was irregular and caused a lot of discrepancy 
from the seasons. 



274 Siddhanta Darpana 

Julius caesar, on his conquest of Egypt in 44 
B.C. was advised by Egyptism astronmer Sosigenes 
that mean length of year should be jlS-i/l 
days.Normal length should be 365 days and one 
extra day should be added every fourth year. The*! 
the fifth month from March, Quintilis was changed 
to July (Julius) in 44 B.C. in honour of Julius Caeser 
and length of months were fixed at their present 
duration. Extra leap year was obtained by repeating 
the sixth day before kalends (first day) of March. 
In 8 B.C., sixth month after March, Sextilis .was 
changed to August in honour of Augustus, 
successor to Caesar. To correct the seasons, §Q days 
were added to 46 B.C. 23 days after February and 
67 days between November and December. Thfe 
year of 445 days was known as year of confusion. 
Caesar wanted to start the new year on 25th 
December, the winter solstice day. But people 
resisted, because new moon was due on January 
1,45 B.C. Caesar had to accept the traditional 
landmark of the year. 

Weekdays of 7 days week were introduced 
sometimes in 1st century AD on pattern of chaldean 
astronomers. Days of crucification of christ and his 
ascending to heaven was fixed arbitrarily on Friday 
and Sunday later on. New Testament only says 
that he was crucified on a day before Passoyttr 
festival of Hebrews which was on full moon day 
of the month of Nissan. 

The present christian era started at about 530 
AD. When era beginning was fixed from the birfli 
year of christ, birth day of christ was fixed oil 
December 25, which was winter solstice day an!8 
ceremonial birth day of Persian god Mithra in 1st 



Corrections to Moon 275 

century B.C. However, a Roman inscription at 
Ankara shows that king Herod of Bible who had 
ordered massacre of children after birth of christ, 
was dead for 4 years at 1 AD. Therefore, christ 
must have been born before 4 B.C. 

The Julian year of 365.25 days was longer 
than the true year of 365.2422 days by 0.00788 
days, so the winter solstice day which fell on 21 
December in 323 AD, fell back by 10 days in 1582 
AD. In 1572, Pope called a meeting to discuss the 
correction. In 1582 Pope Gregory XIII, published a 
bill instituting a revised calender. Friday, October 
5 of that year was to be counted as Friday, October 
15. The century years which were not divisible by 
4 were not to count as leap years. Thus the number 
of leap years in 400 years was reduced from 100 
to 97. length of years was 365.2425 days, the error 
being only one day in 3300 years. This was adopted 
immediately by the catholic states of Europe. But 
Britain adopted it in 1752, China in 1912, Russia 
in 1918, Greece in 1924 and Turkey in 1927. Revised 
rules for easter have not been adopted by the 
Greek Orthodox church. 

World calender : To remove the working 
defects of Gregorian calender, a world calender 
was propsoed to UNO in Geneva meetign of 
ECOSOC in 1954. In this calender week days of 
every year are same. One extra week day in 365 
days is kept after 30th December called W or world 
holiday. In leap year another world day was to be 
introduced after 30th June. Every year was same 
for counting of week days. Each quarters of 3 
Months was of 91 days, 13 weeks. First month of 
each quarter was 31 days and remaining of 30 



276 Siddhanta Darpm 

days. So each quarter has same form of calender! 
Each year (each quarter also) begins on sunday| 
Each month has 26 working days, plus Sundays. J 

(8) Luni Solar Calenders - 

We need very accurate measurements and] 
complicated procedure to tally lunar and solarj 
calenders. Mean lunar synodic month = 29.53058*1 
days 

= 29 d 12h 44 m 2s/ 

with a variation of ± 7 hours 

Mean sidereal period of moon = 27.3216611 
days 

= 27d 7 h 43m 11.5s. 

1 

with a variation of ± 3— hours. 

2 

12 lunations (synodic) amount to 354.36706 
days while tropical solar year is 365.24220 days. 
Length of lunar year is shorter by 10.87514 days, 
and there are 12.36827 lunar months in a solar 
year. Tropical solar year is varying very slowly and 
is becomig shorter by 8.6 seconds == .0001 days 
in 1600 years. Thus at kali beginning or in Sumerian 
times it was 365.2422 days. 

All ancient nations had almost accurate 
knowledge of the mean synodic month. However, 
no rules could be fixed for tallying the lunar year 
with solar year. Hammurabi (1800 B.C.), law giver 
king of Babylonia, has a record saying that the 
thirteenth (extra) month was proclaimed by royal 
order throughout the empire on advice of priests. 
Practically the start of first month was adjusted 
with ripening of wheat. 



Corrections to Moon 277 

* 

Later Babylonians, called Chaldeans around 
600 B.C. fixed some empirical relations in lunar 
and solar years for correction of calender in form 

m lunar months = n solar years. 

where m and n are integers 

Some convenient periods were 

Octaeteris - 8 tropical years = 2921.94 days 

99 lunar months = 2923.53 days. 

This gave 3 intercalary months in 8 years with 
error of only 1.59 days. 

In about 500 BC (383 B.C. according to father 
Kugler) 19 year or Saros cycle was used with 7 
intercalary months 

19 solar years = 6939.60 days 

235 lunar months 6939.69 days 

This gives a discrepancy of 0.09 days in 19 
years or of 1 day in 209 years. 

Their 19 years cycle was of 6940 days with 
leap years on 1st, 4th, 7th, 9th, 12th and 15th year 
in the first month and in 18th year at 7th month. 

First month started with 30 days, then other 
months were atternately 29 and 30 days. Thus a 
normal year was of 354 days, but in 5 years of 19 
year era one extra day was added to last month, 
making the year of 355 days. After adding 
intercalary year, the year was of 354, 355, 383 or 
384 days duration. Effect of this arrangement was 
that the first month Nisannu start was never more 
than 30 days away from vernal equinox. The 
Chaldeans used gnomon for ascertaining time of 2 
e quinoxes and 2 solstices which divide a solar year 
mto 4 almost equal seasons. 



278 Siddhanta Darpam 

Eras of Western world - Dated records of 
kings in Babylon beings from about 1700 B.C. 
(Kassite kings). In Egypt also regnal years were 
used. But in Babylon, months and dates were of 
lunar month is while they were solar in Egypt. 

Hipparchus (140 BC) and Ptolemy (150 AD) 
of Greece used the records of systematic observa- 
tions of Babylone from 747 B.C. since the time of 
one king ftabu Nazir. Though they counted the 
astronomical era from 26 Feb. 747 B.C. in that 
reign, they adopted Egyptian solar years of 365 
days each for ease in calculation of dates. 

Macedonian Greek had their own months, but 
after they settled in Babylon in 313 B.C, they 
adopted their months to Chaldean months, 1st 
month Dios starting with 7th month of Chaldeans 
at autumnal equinox. 

Seleucus, a general of Alexander, a, 

Macedonian Greek founded a big empire in west 

Asia and started his own era Seleucidean era. Iri 

official or Macedonian reckoning it started from the 

lunar month of Dios near autumnal equinox in 

(-311 AD) or 312 B.C, with greak month names. 

In Babylonian reckoning, the months had Chaldeari 

names starting from Nisan near vernal equniox. 

Parthian era was started in 248 B.C. when Persia] 

again became independent empire. | 

Ancient Jewish calender was lunar and theii| 

month names are derived from Chaldean narnei 

or vice versa. The day began in evening an<| 

probably at sunset. Extra month was added whe^ 

necessary by making two months of the last montt| 

Adar - original was named veadar followed bj$ 









Corrections to Moon 279 

Adar. Year beginning was changed from Nisan 
month to Tisri corresponding to Mecedonian month 
of Dios. Around 4th century A.D. rules were 
formed for intercalation. In a cycle of 19 years 
3,6,8,11,14,17 and 19th years had extra month. Start 
of first months was adjusted, so that week days 
of important festivals do not change. Thus a 
common year could have 353, 354 or 355 days and 
a leap year of 383,384 or 385 days. 10 of the middle 
months had got fixed duration of 29 or 30 days. 
Extra month was of 30 days. The other two (1st 
and 12th months) varied according to length of the 
year. Jewish era is called Anno Mundi or libriath 
olum or Era of Creation or Freedom. 

According to mnemonic Beharad, this era* is 
supposed to begin at the beginning of lunar cycle 
on the night between Sunday and Monday, Oct 
7, 3761 B.C., at 11 hours 11-1/3 minutes PM. (Be 
= Beth i.e. 2nd day of week), ha (he = five, i.e. 
fifth hour after sunset) and Rad (Resh) delet i.e. 
204 minims after the hour, 18 minim = 1 minute) 

In Bible, eras have been mentioned from 
flood, exodus, the earthquake in the days of king 
Uzziah, the regnal years of monarchs and 
Babylonian exile. After exile, they counted years 
from Persian kings, and then from Seleucid era. 
Days have also been counted from fall of the second 
temple. 

312 - Seleucidean era = Christian era B.C. (Jan 
to Sept) 

Saleucidean era - 311 = Christian era AD (Jan 
to Sept) 

Year 1 after destruction of second temple 



280 Siddhanta Darpana 

= 3831 Anno Mundi 

= 383 Seleucid = 71 A.D. 

Islamic Calendar - 

This is purely lunar calender now and has no 
connection with solar year. The year consists of 12 
lunar months; beginning of each month is deter- 
mined by 1st observation of crescent moon in the 
evening sky. The months have 29 or 30 days and 
the year 354 or 355 days. The new year day of 
Islamic calendar loses about 1 month in 3 years, 
and completes the retrograde cycle of seasons in 

1 , 
32 — solar years. 

Hejira (A.H.) was introduced by caliph Umar 
about 638-639 AD, stating from evening of 622 AD, 
July 15, Thursday (Since sunset Friday started in 
Islamic calender). Then crescent moon of the 1st 
month Muharram was first visible. This was the 
new year day preceding the emigration of Muham- 
mad from Mecca (about Sept 20, 622 AD.). The 
months are alternately of 30 and 29 days from 1st 
month. Last month is 29 days in normal year and 
30 days in a leap year. If Hejira year is divided 
by 30 and remainder is 2,5,7,10,13,16,18,21,24,26 
or 29 then it is a leap year. Thus 11 leap years in 
30 years, gives the cycle of 10,631 days which is 
0.012 days less than the true value. 

Dr. Hashim Amir Ali of Osmania University 
has showed that the mohamadan calender was 
originally luni-solar. Upto the last year of the life 
of Mohamad; i.e. upto AH 10 or 632 AD, a 
thirteenth month was intercalated when necessary. 
The family of astronomers, known as Qalamas 



Corrections to Moon 



281 



decided at hajj in last month, whether 13th month 
will be added or not. This should have been 3 
times in 8 years or 7 times in 19 years, but use 
of discretion by eldest Qalama created confusion 
afterwards. Thus AH 11, a normal year started on 
29th March 632 AD. after vernal equinox. Thus all 
the previous years with intercalation, started after 
sighting new moon after vernal equinox. Thus the 
initial epoch of Hejira era was at the evening of 
March 19,622 AD, Friday, the day following the 
vernal equinox. 

Names of Lunar Months 



Indian 


Chaldean 




Mecedonian 


Jewish 


Islamic 


Caitra 


Addaru 




Xanthicos 


— 


— 


Vaisakha 


Nisannu 


(30) 


Artemesios 


Nissan 


Muharram 


Jyestha 


Airu 


(29) 


Daisies 


lyyar 


Safar 


Asadha 

4 t 


Sivannu 


(30) 


Panemos 


Sivan 


Rabi-ul-awwal 


Sravana 


Duzu 


(29) 


Loios 


Tammuz 


Rabi-uls-sani 


Bhadra 


Abu 


(30) 


Gorpiaios 


Ab 


Jamada alawwal 


Asvina 


Ululu 


(29) 


Hyperberetrios 


Ellul 


Jamada as sard 


Karttika 


Tasritu 


(30) 


Dios 


Tisn 


Rajab 


Margasirsa 


Arah/Samnah(29) 


Appelaios 


Marheshvan 


Shaban 




Kisilibu 


(30) 


Audinaios 


Kisilev 


Ramadan 


Pausa 

• 


Dhabitu 


(29) 


Peritios 


Tebeth 


Shawal 


Magha 


Shabat 


(30) 


Dystros 


Shebat 


zil kada 


Phalguna 


Addaru 


(29) 


Xanthicos 


Adar and 


Zil hijja 


Caitra 








Veadar 





(9) Old Indian Calendars : 

A. Vedic Calender - Vedic calender was luni 
solar. Year was named in three manners - Solar 
year, civil year and lunar year (normal and 
intercalary). 

Sama = Fixed year or constant. It is opposite 
to 'masa' i.e. formal of 12 masa of 30 days each. 
Thus it means a year of 360 civil days or 365 solar 
days (i.e. 365-1/4 days) 



252 Siddhdnta Darpana 

Lunar years are called vatsara - which are of 
5 types- Samvatsara, anuvatsara, Parivatsara, Id- 
vatsara and Idavatsara. Anuvatsara is also called 
Iduvatsara. When these indicate a sequence of 5 
solar years of 366 days each, vatsara is a sixth year 
of 360 civil days or sama (as per yajus jyotisa). 

Names of thirteen months in Taittiriya 
Brahmana (3-10-1) are Aruiia, Aruna rajas, 
Pundarika, Visvajlta, Abhijit, Ardra, Pinvamana, 
Annavan, Rasavan, Ira van, Sarvan sadha, Sambhar 
and Mahasvan, Mahasvan appears to be increased 
month (with extra days in a solar year). 

6 seasons of two solar months each are as 
follows - 

1. Vasanta - Madhu and Madhava 

2. Grisma - Sukra and Suci 

* 

3. Varsa - Nabhas and Nabhasya 

4. Sarad - Isa and urja 

5. Hemanta - Sahas and Sahasya 

6. Sisira - Tapas and Tapasya 

Taittiriya Brahmana has given a list of 24 half 
months (1 fortnight), names of day times and night 
times in sukla and krsna paksas - 60 names, names 
of 15 muhurtas in sukla paksa day and night, 
Krsna paksa day and night - 60 names and 15 
parts of each muhurta / called prati muhurta). 

Name of lunar months were named after the 
naksatras entered by moon on purnima day. 
Rk veda (1-15-1) tells that^Indra drinks soma juice 
with seasonal adityas on full moon day. Thus Indra 
is always at a point 180° away from sun. 



Corrections td Moon 2S ^ 

Aditya corresponding to different seasons 

are 

(1) Mitra - sisira (2) Aryaman - vasanta (3) 
Bhaga - grisma (4) Varuna - Varsa (5) Daksa or 
Dhata - sarad (6) Arhsa - Hemanta 

Rk veda verse 10-72-4 by Sunahsepa gives 
method of deciding about inclusion of intercalary 
month - 

Daksa was born of Aditi and Aditi was Daksa's 
child. The whole ecliptic was Aditi and its division 
were adityas - 6 for each season, 12 for each month 
or 13th for extra month. First point of Daksa 
division was the start of ecliptic zero degree. Year 
started with rise of this point on eastern horizon 
with sun. When the next rise was not before 13th 
full moon, 13th month was extra month otherwise 
it was month of next year. In santipatha also it is 
stated — 

In Vajasaneyi samhita, two adhika masa are 
named. Sansarpa is extra month before winter 
solstice. Another is malimluca. Ksayamasa (lost 
month) was' called Arhhaspati. (Yajur-VS, 22-30) 

In a solar year of 365-1/4 days, 5 or occasional- 
ly six days are extra after civil year of 360 days. 
These have been called atiratra (i.e. extra days after 
grand night). Taittiriya Samhita (7-1-10) says that 
4 atiratra make the year incomplete, while 6 atiratra 
give excess, so five are the best. 

Aitareya Brahmana has denned Tithi as the 
time during which Moon sets and rises again 
(32-10). Thus like civil day from sunrise to sunrise, 
tithi is a moon day from moon rise to moon rise. 



284 

Siddkanta Darpana 

r. 

In sukla paksa tithi was from moon set to 
moon set, and in other it was moon rise to moon 
rise. 

Atharva Vedanga jyotisa has defined two 
karana in each tithi - one from moon rise to moon 
set and second from moon set to moon rise. These 
tithi and karana were of unequal length. Later on 
they were made of equal length defined on basis 
of moon phase. 

B. Vedanga Jyotisa (Rk veda) : This is 
described in only 36 verses in anustupa chanda 
including introduction and importance*. This is one 
of the six parts of Rk veda. Though it is shortest, 
it gives a most comprehensive, luni solar calendar 
so far. It was written by 'Lagagha' whose place 
was 35° N latitude, nothern border of Kashmira, 
may be the present town of Alma-Ata of Kyrghiz! 
This place might have been first place of learning^ 
hence first school is called alma-meter. 

Efforts to explain its meaning on basis of 5 
years cycle (yuga) were unsuccessful, by various 
anthors as B.G. Tilak, S. B. Diksita and T.S.K. 
Shastri, R. Shamshastri etc. Tanca samvatsara 
mayam yugam' was interprated that a yuga has 5 
years (meaning of samvatsara). But samvatsara is 
one of the 5 types of lunar years and its meaning 
should be - A yuga has 5 years of samvatsara type, 
remaining years of other 4 types. If calculations 
are made on that basis, a yuga has 19 years, with 
5 types of vatsaras, out of which 5 are sanvatsaras. 
This also gives meaning of other types of years. 
This gives correspondance of solar and lunar years 
in terms of tithis, days and naksatras also. 



Corrections to Moon 285 

Time cycle : There are 360 tithis in a lunar 
year. Solar year is bigger by 10.89 days. With 
reasonable accuracy, 7 intercalary months 
(adhikamasa) occur in a cycle of 19 years. Thus 

228 solar months = 235 lunar months Addi- 
tional 7 months form 7 x 30 = 210 tithis. Thus 

there is difference of — — = 11— tithis (10.89 days) 

19 19 v J ' 

between a solar and a lunar year. Thus a solar 

year consists of 371 — tithis. If we assume a leap 

year in a cycle, we have 18 years with 371 tithis 
and one year (i.e leap year) with 372 tithis. This 
cycle of 19 years is called a yuga. 

Calculation of Rtu sesa - A year has 12 months 
(lunar) each having two parts sukla paksa (called, 
sudi - i.e. Suksma diwas - or suddha diwas) and 
Krsna paksa (Badi i.e. bahula diwas, extra days). 
Thus 24 paksa of a year have difference of 11 tithis 

from solar year. Difference in each paksa is — = 

0.458 = — tithi approximately 

Thus a lunar paksa = 15 tithis 

Solar paksa = 15 — tithis 

For calculating extra tithis for each half of 
solar year, we have to add 11/2 tithis = 11 karanas. 
Thus we have to subs tract 1 karana after half year 
from 12 karanas got after taking 1 karana for each 
paksa approximately. One tithi is taken extra in a 
completed yuga of 19 years. When cumulative total 
of extra karana after a semester is more than 60 



286 * Siddhanta Darpana 

karanas = 30 tithi = 1 month, one extra month is 
added in that semester. 

Classification of years : If one karana is not 
dropped in each semester, then Rtusesa will be 12 
tithis per year or 6 tithis per ayana (semester). 
Thus start of ayanas will be after 6 tithis each and 
after 5 ayanas (2-1/2 years) the cycle will be 
complete and one extra month will be added so 
that the month starts again with Magna sukla 1 
(1st year of month). Thus years were classified 
according to range of tithis on which 1st day of 
year fell - 

Samvatsara - Sukla 1 to 6th 
Anuvatsara - Sukla 7th to 12th 
Parivatsara - Sakla 13th to 18th (or badi 3rd) 
Idvatsara - Badi 4th* to 9th 
Ida vatsara - Badi 10th to 15th 
When we decide adhikamasa for each lump 
of 60 karana Rtusesa, 5 years in 19 years yuga are 
of samvatsara type. 3 years are Idavatsara, lagging 
behind most, hence the adhikamasa occurs in 1st 
semester of those years (6th, 9th and 17th). which 
can be seen by calculation. Four years are of 
Idvatsara type lagging 18-24 tithis, hence the adhika 
masa is added in 2nd semester of (3rd, 11th, 14th 
and 19th years) 

Naksatra calculation for sidereal lunar year - 

In a lunar month, moon completes its circular 

journey of 27 naksatras and travels about 2 

naksatras more. To be more accurate, it completes 

13 revolutions in 12 lunar months. Thus in 19 solar 



Corrections to Moon 287 

years, there are 254 sidereal months and in 19 
lunar year 19X13 = 247 sidereal months of moon. 

Thus difference in 2 cycles is 7 sidereal months 
= 7 x 27 = 189 naksatras. Thus we have 10 naksatras 
per year for 18 years and 9 extra naksatra in one 
year 

1 solar year = 361 naksatra 

Leap year = 360 naksatras 

1 lunar year == 351 naksatras ( = 13 x 27) 

10 
Solar semester - lunar semester = — = 5 

naksatras. 

Rtuses in terms of naksatras is calculated by 
assuming a total of 190 naksatra for 38 semester 

i.e. 5 for each. To be more accurate it is 5- — 

naksatra. Thus calculation is to be started from 
sravistha (Sravana + 38 parts of l/38th i.e. complete 
dhanistha). Complete naksatra divisions will be 
counted from Sravana at interval of 5 naksatras 
each for every ayana (semester). Thus list of 
naksatras is given at intervals of 5 naksatras 
indicating only one letter from each naksatra. This 
verse was deciphered brilliantly by Sri S.B. Diksita. 

According to moon naksatra at start of year 
also years can be classified. Thus samvatsara can 
start from sravana upto 5.4 naksatra (Asvinl) - 2.7 
difference for each semester. Anuvatsara is Asvini 
(5.4) to Ardra (10.8) - countings done from sravana. 
Parivatsara can start up to 16.2 (uttaraphalguni) 
Idvatsara upto (21.6) anuradha and Idavatsara in 
remaining naksatras. 

C. Visvamitra's astronomy - also indicates a 
19 year yuga. His hymn in Rk veda III - 9-9 reajds 



'■■-31 



4 



288 Siddhanta Darpana 

i.e. 3339 devas (dyues or parts of ecliptic = 
aditi) worshipped agni (sun or krttika naksatra) by 
rotations in the sky. 

This is based on calculation of solar naksatras 
for each parva (or paksa). First solar year of 372 
tithis is the basic year in which sun crosses the 27 
naksatras. Thus 1 solar naksatra = 372/27 = 124/9 
tithis. To avoid fractions, angular distance travelled i 
by sun in a tithi is divided into 9 parts called 5 
'Bha-amsas' or bhansa. (Bh°) 

1 Tithi = 9 Bh° of sun f 

1 Naksatra = 124 Bh ° j 

13° 20' 1 

1 Bh° = = 6'27".l approx | 

1 parva = 15 tithi = 135 Bh° = 1 Naksatra + j 
11 Bh° 

In 1 Ayana = 12 parva i 

= 12 (Naksatra + 11 Bh°) = 12 Naksatra + 132 1 
Bh ° 

= 13 Naksatra + 8 Bh° 

Thus 8 Bh° arise in one parva (in addition to j 
completed naksatras). 

In 371 tithis of a solar year there are 371x9 * 
Bh° = 3339 Bhansas, which are indicated by 
Visvamitra. 

Saros cycle of Chaldea was of 18 years and 
10.5 days after which eclipses are repeated. 

3339 tithis = 111 synodic months + 9 tithis. 



Corrections to Moon 289 

This is half of Saros cycle of 223 synodic 
months 



3339 



/^.n \ 



synodic years = 



3240 



1080 



= 1080 



3 

sideral years are a yuga x (1/3 of Visvamitra's 
mahayuga). This is the period of precession of 
equinoxes for 1 parva (15 tithi or 15° movement 
of sun in 15 x 72 = 1080 years). 

A day is divided into 603 kalas = 30 muhurta. 
Moon crosses 1 naksatra in 610 Kalas. 

D. Yajus Jyotisa : This is part of Yajurveda 
whose commentary by Somakara was available. It 
has 44 verses out of which 30 are common with 
Rk jyotisa Due to that reason, scholars have tried 
to combine these two into one text of 50 verses 
and interprate both on basis of 5 years yuga. 
However, Rk has 19 years yuga and yajus has 5 
years yuga. This has been specified in verse 31 of 
this text .-'.-. 

(In a yuga) there are 61 savana months, 62 
lunations and 67 sidereal months i.e. naksatra 
masas; 30 days make one savana month and 30-1/2 
days make one solar month. This along with verse 
4 tells that we have one adhika masa after every 
2-1/2 years - clearly specify a 5 year yuga. This is 
the different meaning of this text. 

Adjustment of luni-solar years in this system 
can be done on basis of two statements in 
contemporary texts. Mahabharata, santi parva ch 
301 tells. 

SR «'<4<tK]U|i I ^*TOHf ^ m tot 

(By way of leap) drop years as well as months. 



290 Siddhanta Darpan 

Order of leap years is indicated by Taittirlya 
Brahmana (part of yajurveda) (3-10-4) which gives 
list of years. 

(1) Sarhvatsara (2) Parivatsara (3) Iduvatsara 
which is same as anuvatsara according to Madhava 
(4) Idvatsara (5) Idavatsara and (6) Vatsara : 

This extra sixth vatsara is a sama or savana 
year of 12 savana months (30 days each according 
to verse 31) and comes after each yuga of 5 years 
in a yuga in the above order. Thus each of the 
five years is of 366 days and sixth year of 360 days 
balance the extra days counted in the yuga. The 
year after sama or vatsara may not be samvatsara, 
it will be decided according to tithi or naksatra at 
beginning of year as defined in Rk jyotisa. Thus 
the ommitted years can be thought as dropped 
years or ksaya years. Thus in a cycle of 5 yuga% 
of 25 years we drop six years including 3 leap 
years whose adhikamasa also gets dropped in the 
process. Thus we get 19 year yuga as before. But 
we get a simpler 5 year yuga. 

There is a difference of 4 hours 23 minutes 
between vedanga cycle of 19 years and 19 solar 
sideral years. It we acid 8 years at the end of eight 
19 years yugas, we get 160 years. The difference 
at that end reduces to 23 minutes only. 

Time period - RVJ verse 5 tells that year 
started on full moon day in Magna in winter season 
when sun was in vasava nakshatra (1st year of 
yuga) 

YVJ verse 6 tells that sun in beginning of 
sravistha indicated beginning of Magna and sun 






<3 



Corrections to Moon 291 

* 

in mid point of aslesa started beginning of south 
solstice. 

Assuming 1 ° precession in 72 years, this 
indicates Rk jyotisa in 2976 BC and yajus jyotisa 
in 2352 BC. However verse 3 of RVJ indicates that 
the theory was coming since long when these 
verses were composed. 

E. Gavlm Ayana has been mentioned in 
Tattiriya samhita, Satapath brahmana, Gopatha 
brahmana and Baudhayana srauta surra. It indicates 
a 4 year yuga with 1 leap year according to Prof. 
R. Shama Sastri (1908 - gavam ayana) Accumulation 
of l/4th day of each of previous 3 years combined 
with 4th year to make one extra day like the Julian 
calender. Thus this is a cow with 4 legs or 3 parents 
of sun. Four years of this yuga were called kali, 
dwapara, treta and krta yuga. These are also called 
1st, 2nd, 3rd and complete (krta). Krta is also called 
Satya or Rta i.e. which really came (as a full day) 
If year or yuga starts in evening, 1st year (kali) 
will end at midnight after 365-1/4 days (sleeping 
time). 2nd year dvapara will end in morning (rising 
time) on 366th day. 3rd year Treta will end on 
366th day noon, when sun is at highest. 4th year 
krta or satya will end in evening when people are 
moving. Thus Aitareya Brahmana tells - sleeping 
is kali, rising is dvapara, standing is treta and 
moving is krta, so keep on moving (7-15). This is 
attributed to Manu. This was around 23,720 B.C. 
as Taittirlya samhita indicates (7-4-8) vasanta at 
phalguna full moon. Rk veda indicates rains in 
Mrgasira naksatra indicating same time. 

Thus 4 years yuga with 4th as leap year 
appears to be first system started around 24000 
B -C. Then 19 years yuga with 7 leap years (lunar) 



292 



Siddhanta Darpaqm 



4 



'M 



W\ 



4 



of vedanga jyotisa continued upto about 3000 B.C 
i.e. kali beginning. With kali era smaller 5 year 
yuga of yajurveda forming a 19 year yuga was 
started. 

F. Jaina calendar - Surya prajnapti and Candra 
prajnapti are two principal texts written at the time 
of Mahavira about 600 B.C. However Jain Tir- 
thankars and their astronomical traditions might 
have started along with yajurveda or early 
brahmana texts. 1 

* '<: 

It 

There are five kinds of sarhvatsaras (years) «• 1 

(1) naksatra samvatsara (2) yuga (cycle) sam-| 
vatsara (3) Pramana (standard) samvatsara (4)| 
Laksana (symptomatic) samvatsara and (5) sanlcara 
(saturn) samvatsara 

Days in year 



Nakstrika 

Lunar 

Rtu 

Solar 

Abhivar- 
dhana 



327-51/67 days 

354-12/62 

360 

360 

382-44/62 



Months 
in year 

27-21/27 

29-32/62 

30 

30-31/62 

31-121/124 



1 



Month 
of 5 year cyde 

67 

62 

61 

60 

57-3/13 



l 



■■„( 



A five year yuga consisted of 5 lunar 
samvatsaras with 3rd and 5th years having extra; 
months called abhivardhan samvatsara. This i$ 
almost like yajus jyotisa but simpler. Naksatra 
samvatsara was named according to naksatra 
occupied by Jupiter at the time of complation of 
samvatsara - these are same as present months 
names of India. Sanlcara samvatsara was time of 
sard in crossing 1 naksatra out of 28 with mean 
motion. 



Corrections to Moon 293 

(10) Indian Eras : 

(A) The eras started after kali era are based 
on concept of Mahayuga of 43,20,000 years or a 
kalpa of 1000 mahayuga. Yuga concept is attributed 
to Aryabhata in 499 AD and kalpa concept to 
Brahmagupta in 627 AD. However, Brahmagupta 
refers to Visnudharmottara Purana. Smrtis also 
have been referred to. Aryabhata himself has 
followed purana tradition, except to treat four parts 
of yuga as equal. The five limbs of calendar, known 
as pancanga have already been explained in the 
previous chapter. Their brief definitions are given 
again - ° 

(1) Vara - Runing weakday in a cycle of seven 
days. 

(2) Naksatra - Naksatra occupied by moon. 
This almost means one day and was most 

popular hi. mahabharata era to indicate a day (in 
Valmiki Ramayana also). 

(3) Tithi - Moon rise to moon rise system was 
changed. Vedanga jyotisa started equal division of 
titrus depending on phases of moon. 

Tith* = M° on - sun 

12° 

pvf. S U ° ti f ntS above 15 indicat e krsna paksa and 
«ctra days beyond 15th are counted as tithi number. 
£ence Krsna tithi is called' bahula divas' (extra 

^u° r ' badr " short Sukla tithi ^ called sudi - 
suddha divasa. 

rko / 4) ****** is half of tithi. In veda it was moon 
se to moon set or vice versa. From vedanga jyotisa 
lt is exactly half of tithi. * W • 



294 Siddhdnta Darpctr0 

Moon - sun 

Karana = — 

6 

A 

Corresponding to 1 day = 2 karanas extra at I 
beginning of 19 yeear vaidika yuga, one karana at | 
each end of amavasya are omitted from running j 
cycle of seven karanas. These 4 are fixed karanas. j 
Remaining 56 karanas start from sukla 1st tithi 2nd| 
half in which seven karanas are repeated 8 times I 
in a month. j 

(5) Yoga - It is only a mathematical concept. 1 
It means sum of longitudes of sun and moon and A 
one cycle of 360° makes 27 yogas. Originally there! 
were only 2 yogas. Vyatipata was when kranti of | 
sun and moon were equal but their longitude was j 
equal and in opposite directions. When longitude '■ 
is same but kranti is equal and opposite, it was ! 
called 'vaidhrti'. Thus yoga was means to calculate" 
these and subsequently others were included to 
make a complete cycle of 27 yogas like 27 naksatras. 

There is another kind of yoga which is 
combination of vara, tithi or naksatra for auspicious 
works. 

B. Rules for calender - 

(1) A lunar month stars from Sukla 1 (called 
ananta) or from Krsna 1 tithi called Purnimanta. 
Lunar month is named after the naksatra ap- 
proximately occupied by moon on purnima of that 
month. Amanta is called mukhya and other gauna. 

(2) A solar month starts with entry of madhya 
surya in a nirayana rasi (i.e. fixed point of zodiac). 
The first day of month may start on same day, 



Corrections to Moon 295 

next day or 3rd day according to occurrance of 
sankranti in different parts of day or night. 

(3) In luni-solar year a lunar year tallies with 
a particular sankranti of surya every year. In a 
lunar month when there is no sankranti the month 
is called adhika masa. The month having two 
sanknantis, is called ksaya masa, the month 
corresponding to 2nd sankranti of the month is 
dropped. Ksaya masa is called amhaspati. The 
adhika masa before it is called sansarpa (or in 1st 
ayana uttarayana of the year). The adhika masa 
after ksaya masa or in 2nd ayana is called 
malimluca. 

(4) Uttarayana starts when sun starts its 
northward journey after winter solstice or sayana 
makara samkranti (24th or 25th December). 
Daksinayana starts when sun starts going south 
from summer solstice i.e. sayana mithuna sankranti 
(26 June). A year may start with start of uttarayana 
as in vedanga jyotisa or from equinox in uttarayana 
- vernal equinox which is middle point of 
uttarayana. Instead of exact equinox point of 
uttarayana, we count entry into fixed zodiac rasi 
which follows 23 days later at present. 

C. Rules for sankranti 

(1) In Orissa, solar month begins on same 
day as sankranti of madhyama surya, irrespective 
of the part of day (sun rise to sun rise) it falls in. 

(2) Tamil rule - If sankranti takes place before 
sunset the solar month begins on same day, 
otherwise from next day. 

(3) Malabara rule - In parahita system of 
Kerala, if sankranti takes place before lapse of 3/5th 



296 Siddhdnta Darpana 

of duration of day (i.e. about 18 ghati or 7h / 12m 
after sunrise - about 1-12 p.m.), month starts on 
the same day, otherwise from next day. 

(4) Bengal rule - When sankranti takes place 
before midnight, month starts from next day, if it 
is after midnight, then from third day (next to next 
day). 

If sankranti is within 1 ghati of mid night, 
i.e. 24 minutes before or after, tithi at sunrise time 
is examined. If sankranti is before lapse of that 
tithi then month starts on next day. If it is after 
tithi then from 3rd day. For karka and makara 
sankranti this rule is not followed. 

D. Eras started in India - 

Eras of long period have been described in 
all old civilisations. They describe three great 
floods. After one great flood Brahma appeared and 
started the civilization. Then saptarsis were born, 
rule of Daitya, Deva and Danava followed. That 
may be called Deva yuga. 

Devayuga ended with another great flood in 
which rudiments of life were preserved by Manu 
(or Nuh). After re-settlement human eras began. 
These were formed into a cycle of 12,000 divya 
years, l/10th was kaliyuga, 2, 3, 4 times were 
dvapara, treta and krtayuga (or satyayuga). 

There was another era called sap tar si era 
which is equal to 2700 divya versa, assuming that 
they remain in one nakSatra for 100 such, years. 
Another count in Vayupurana mentions saptarsi 
yuga as 3030 manusa varsa. Thus divya varsa 
appears to be solar sidereal year of 365-1/4 days 
and manusa varsa is sidereal lunar year of 327.4 



Corrections to Moon 297 

days. These values give the above ratio of values 
given in Vayu pur an a. 

Each yuga was further divided into sub parts, 
like Vayu purana indicates 24 parts of treta yuga 
and great personalities have been named in each 
part. Each part considered equal, parts of Treta 
were of 150 years each. Dvapara had 28 parts = 
2400 solar years = 85.7 years approx* It is more 
convenient to keep 4 parts as sandhi periods after 
treta and after dvapara. Then each part is of 1 
century, i.e. 1 naksatra of saptarsi. Some cor- 
roborating quotations for this time scale are - 

(1) Megasthenese quoted by Pliny (Indika of 
Arian ch IX) - 

From the days of Father Bacchus to Alexander 
the Great, their (Indian) kings are reckoned at 154 
whose reigns extend over 6451 years and 3 months 
(Pliny } 

• Father Bacchus was the first who invaded 
India and was the first of all who triumphed over 
the vanquished Indians. From him to Alexander 
the Great, 6451 years 3 months. ... .reign by 153 
kings in intermediate period (Solin) 

From the time tb Dionyson (or Bacchus) to 
Sandrokottos> the Indians counted 153 kings and 
a period of 6042 years. Among these a republic 
was thrice established, another for 300 and 120 
years. 

Note - Bacchus is mentioned in Bible, becomes 
Dionyson in Greek. This is derived from 
'Danusunu' (son of Danu third wife of Kasyapa — 
or danava) or Vipratitti (Bacchus). Herodotus has 
stated that Bacchus was called Orotol in old Arabic. 
This is derived from Vipratitti. 



* Siddhanta Darpan* 

(2) Among sons of Kasyapa prajapati, eldest 
were born from Diti called Daitya. Next were from 
Aditi called Aditya or deva. Daitya were earlier 
and first to rule over world, hence they were called 
'purvadeva'. Last were born from youngest wife 
Danu called danava. Daitya and danava were 
called asura and they were anti to deva or Sura. 
Herodotus writes on basis of Egyptian priests (part 
1, p. 136) 

The twelve gods were, they affirm, produced 
from the eight and of these twelve Hercules is one. 
Hercules belongs to the second class, which consists 
of twelve gods and Bacchus belongs to the gods 
of the third order (P. 199). 

Note - Hercules is derived from 'sura kulesa' 
i.e. Visnu. One form of Visnu was vamana, 

youngest of the twelve adityas, who conquered 
Bali. n 

Daitya Hiranya kasipu = Zeus 
Prahlada = Epaphos = Libye 
Virocana = Beor 

Bali = Bala = Bel = Baalim 

I — 



y 



Bana Candrama = Cadmus 

Greek names are according to Pedigree by 
Nounos (1-377), Bible Duternomy 23-4 tells - they 
hired against thee Balaam the son of Beor of Pethor 
of Mesopotamia. In Jesus 3/7 and 6/28,30, it is 
called Baalum and Baal. 



Corrections to Moon 299 

Devayuga before krta yuga has been men- 
tioned in Ramayana Bala kanda 9/2 and Jaimini 
Brahmana 2/75, Mahabharata Adiparva 14/5 
Sabhaparva 11/1, Vanaparva 92/7. 

(3) Floods - Encyclopaedia of Religion and 
Ethics - 

Article on ages - The cuneiform texts mention 
kings before the flood in opposition to kings after 
the flood. In times before the flood, there lived 
the heroes, who (Gilgames Epic) well in the under 
world, or like the Babylonian Noah, are removed 
into the heavenly world. At that time, there lived, 
too, the (seven) sages. 

Berosus, priest of Marduk temple of Babylon 
under rule of Selucus writes - There were 86 kings 
after flood in first family who ruled for 34,090 
years. Then 5 more families ruled one after the 
other. 

It is note worthy that among the south 
Amercian Indians, it is generally held that the 
world has already been destroyed twice, once by 
fire and again by flood; as among the eastern 
Tupies and Aravaks of Guiana. 

Saving of civilisation from flood in a great 
boat has been described in south America also - 
Tales of Cochiti Indians - Bureau of American 
Ethnology. Bulletin 98 page 2-3. 

(4) Herodotus writes on basis of Egyptian 
calculations - (part 1 page 189) 

Seventeen thousand years (from the birth of 
Hercules) passed before the reign of Amasis. And 
even from Bacchus, youngest of the three, they 
count fifteen thousand years. 



300 Siddhdnta Darpana 

Vayupurana tells 12 deva in 1st treta yuga. 
For further material - Bharata varsa ka Brhat Itihasa 
- by Bhagavaddatta, Pranava Prakasan, Delhi - 26 
may be referred. 

E. Eras since Kali (i) King Yudhisthira 
ascended throne after Mahabharata war and the 
time since then is counted as Yudhisthira saka. 

• * 

Varahamihira writes in Brhatsamhita that according 
to old Garga, Saptarsi were in Magha during the 
reign of Yudhisthira. 36 years after that kali era 
started with death of Krsna 

• • * 

(2) Kali era - It started 36 years after 
Mahabharata war on the day Krsna died. After 
some months Yudhisthira relinquished his throne. 
According to Alberuni (part 3 p. 239), it started on 
13th tithi of Asvina. There are thousands of 
documents mentioning this era. All text books 
count the day from kali beginning. Accordingly, it 
starts on 17/18-2-3102 B.C. Ujjain midnight on 
Friday. Its months are both Caitradi (luni solar) 
and mesadi (solar). Year expired in kali era is 
obtained by adding 3179 to saka year. 

(3) Saptarsi era - It was also called laukika 
kala and Sastra kala in Rajatarangini where it has 
been followed as the standard. This era began on 
Caitra sukla 1st tithi in kali year 27. Saptarsis remain 
for 100 years in one naksatra and after each century 
in a naksatra the years are counted afresh. Era is 
mentioned merely by the naksatra name in which 
saptarsi remain. Thus years are found by adding 
46 to saka era, neglecting the centuries. 

(4) Old Saka era of Varahamihira - Brhat- 
samhita 13/3 tells that Saka starts 2526 years after 



Corrections to Moon 3G1 

king Yudhisthira. This is 554 before Vikrama era. 
According to this saka era, time of Varahamihira 
was 427, i.e. 127 years before Vikrama era. (epoch 
of Pancasiddhantika) He himself writes in kutuhala 
manjari that he was in beginning of Vikrama 
samvat. Traditionally he is reputed to be one of 9 
wise men with Vikramaditya who started Vikrama 
samvat. Such a great astronomer was needed to 
start the new era since Vikrama, Sri S.B. Dlksita 
also has stated in his history of Indian Astronomy 
vol II. p. 2 that the siddhantas mentioned in 
pancasiddhantika belong to 5th century before Saka 
era (new). Thus his epoch of 427 old saka is a 
convenient period about 100 years before Vikrama, 
not his time. The subject matter also is much older 
than Aryabhata during whose time the older, 
theories were extinct. 

(5) Sudraka or Sri Harsa samvat - (2644 Kali) 
Sudraka was also called Sri Harsa who was a king 
of Andhra Kula. Albiruni (chapter 49) has written 
that Sri Harsa was 400 years before Vikrama. 

Ain - Akbari (description of UjjainI) tells that 
difference between Aditya Ponwara (Sudraka) and 
Vikramaditya of Vikrama era was 422 years. 

Yalla in his Jyotisa Darpana (Saka new 1307) 
has written hi«»Pm U u I <^H1 (2345 or 2645) %#M<\: ^rf<m 
Taking 2645 as correct version, Sudraka era started 
in kali 2645 or 399 years before Vikrama. 

This Sudraka has written 'Mrcchakatikam' a 
famous drama. He ruled over Malwa, Kannauja, 
Kasmira etc. After 400 years, 2nd Vikram samvat 
became more popular and this era was forgotten. 



302 Siddhanta Darpang 

This samvat was also called Krta samvat. King 
Samudragupta has written that (Krsna Carita) 

'His rule was rule of law and religious, Hence 
his era was caled Krta samvat/ 

This was written as Malava samvat because 
of its start in Malava. 

Jain Acarya Hemacandra also has mentioned 
that rule of Sudraka was famous for righteousness 
(Kavyanusasana - Bombay edition p. 464). 

(6) Parad samvat - This is Indian name of 
Parthian era or Arsacid era starting in 246 B.C. in 
Iran. This was in use in West India. 

(7) Vikrama Samvat (kali 3044) - This was also 
called Sahasanka year. All Gupta kings used 
Vikram name, so this is connected with one of 
them. In old geneology, Samudragupta is con- 
sidered 93 years after Vikramaditya of Avanti (or 
Ujjain). In north India, it is Caitradi with 
purnimanta months. In Gujarat it is kartikadi and 
months are amanta. Jain inscriptions have called it 
Gupta era also. 

(8) Christian era started with British rule in 
India. 

(9) Salivahana Saka (78 A.D. - Alberuni has 
written (part-3) - One Saka king ruled in areas 
around Sindha river and through his tyranny he 
tried to destroy the Hindu culture. He was either 
a sudra of north west border or from a western 
foreign country. In the end, one king from east 
came and expelled him. After killing Saka king he 
was called Vikramaditya and another era started 
with him. 



m 



"A 



Corrections to Moon 303 

This Vikramaditya might have been Skan- 
dagupta according to purana chronology. 

This is most popular, among astronomers. 
Amaraja Brahmagupta, Bhaskara II, have written 
that, 3179 years of kali had passed at the end of 
Saka king. 

Alberuni tells that Gupta - Ballabha samvat 
started 241 years after Saka. Gupta empire lasted 
for 242 years. Thus Gupta empire and Saka kala 
started together. 

Mesadi solar years are followed in Tamil and 
Bengal and caitradi lunar year is followed else- 
where. Lunar months are purnimanta in north 
India and Amanta in south India. 

(10) Kalchuri era or Cedi era : Kings of Bhoja 
kula ruled in Cedi (present Bundelkhanda) 

Yallaya in Jyotisa darpana has written that 

Bhojaraja samvat = Saka year + 50 

According to this, it started in 28 AD or 85 
years after Vikrama era. 

Keelhorn assumes it to start in 255 AD. 
considering Narasinha deo of Kalinga and of Dahal 
(M.P.) as same person. This year started from 
'Asvina Sukla 1. 

(11) Valabhi era - One Vallabha ended the 
rule of last Gupta king who was a tyrant and 
started ths era in Saka 242. (This has already been 
mentioned in statement of Alberuni under para 9 
above). 

Vallabhi king §uaditya had dispute with a 
merchant Ranka of his town. Ranka invited Hindu 
^ng Hammira of Gajani (Afganistan). In a night 



304 Siddhdnta Darpat 




raid Ballabhi was distroyed in Vikrama era 35 which 
was also referred. ■ 



m 
■•.« 



(12) Hijri Era - This started with Islamic rule 
in India. 

(13) Kollam or Parasurama Era - This is knowri? 
as Kollam (western) Andu (year). This is used irl 
Kerala and in Tirunelveli district. This is sidereal 
solar year starting from solar month of Kanya in 
north Mahabar and simha month in south. This 
year runs in cycle of 1,000 years and present cycle 
is said to be fourth. It's 4th cycle started in Saka 
747 or 824 A.D. According to Mahabharata, 
Parasuram was in sandhi of treta and dvapara (i.e. 
24th part). Thus Parasurama must have been above 
5000 years before 824 AD, may be 6000 completed 
years. 

(14) Nevara year started in 878 AD, with 
Karttikadi amanta months. It was used in Nepal 
upto 1768 AD. 

(15) Calukya Era - Calukya king Tribhuvana 
Malla started this era in 997 AD. 

(16) Simha Samvat - It was started in Gujrat 
in 1170 AD. Months are amanta and start with 
Asadha. 

• • 

(17) Bangali Fasall etc - 

Bangali san started in 593 AD. It is solar year 
and 1st month starting from mesa sarhkranti is 
called Vaisakha (it is called caitra elsewhere). All 
month names are lunar. 

Vilayati san started previous year with Kanya 
sankranti i.e. 7 months before Bangali san. The 
year is solar with lunar month names. This was 



Corrections to Moon 305 

used in Orissa. Difference in rules of sankranti has 
already been explained. Amli Era also was used in 
Orissa with luni solar months. This year started 
from Bhadra sukla 12th (the month of kanya 
sankranti), which is supposed to be birthday of 
kings Indradyumna of Orissa, in purana era. 

Fasali san was started by Akbar. This started 
with same year number as Hizri era but it was 
solar calender to tally with harvesting time. In 
north India, it started in 1556 AD with Hijri year 
963. In south India it started in 1636 AD when 
Hijri year had become 1046. Thus years in South 
India are 2 more. In north India, Fasali year started 
from Asvina Krsna 1 purnimanta. Then it was luni- 
solar. In Madras, it started with karka sankranti. 
The initial date was fixed by British on 13th July 
in 1800 AD and from 1st July in 1855 A.D. 

(18) Laksmanasena Era - It is current in MitMa 
region of north Bihar. This is Karttikadi, amanta 
and started in 1118 A.D. 

(19) Raja Saka - It started on Jyestha sukla 
13th in Saka 1596 with coronation of Sivaji. 

(20) Ilahl Era - This was started by Akbar and 
also called Akbar san. It started on 14-2 - 1956 
A.D. with his coronation. Its years and months 
are solar. Month names were Persian starting with 
Farvardin and each day of month had a separate 
name as in Persia. 

(11) Festivals and Yogas in India 

A. Rules Festivals are generally based on tithis 
except sankranti days. As a tithi generally covers 
a period of two days, a tithi may be counted on 
day when it is current on sunrise. But for religious 
purposes it may have to be celebrated on the 
previous day when it begins. Tithi for feast or fast 



306 Siddhanta Darpt 

is observed on the day in which it covers the 

prescribed part. f 

For such purposes, a day is divided into 5't 

parts between sun rise and sun set - j 

(a) Pratah Kala - 6 ghatika from sunrise. J 

(b) Sarhjava - 6 to 12 ghatika from sunrise. J 

(c) Madhyahna - 12 to 18 ghatika from sunrise, j 

(d) Aparahna - 18 to 24 ghatika from sunrise. ; 

(e) Sayahna - 24 to 30 ghatikas from sunrise. \ 
Relevant parts of night are - 

(a) 4 ghatikas before sunrise are called 
arunodaya or usakala 

(b) 6 ghatika after sunset are called pradosa 

(c) 2 ghatika in middle of the night are called' j 
nisitha - midnight 

A tithi is purva viddha when it commences? 
more than 4 ghatikas before sunset of one day and; 
ends before sun set of the following day. A festival > 
on such a tithi is calebrated on the first day of| 
the tithi and not on the second. 

Tithi dvayam - when 2 tithis meet between? 
18 and 24 ghatikas after sunrise, but a similar- 
meeting does not take place on next day. 

B. Festivals Connected with naksatras as well 
as Tithis 

In southern India, naksatras are often linked 
with solar months to observe a festival. Sravistha 
with Lunar sravana makes upakarma. Tithi festivals 
are also connected with solar months. When a 
sukla pakssa tithi falls twice in a solar month, the 
first is called a sunya tithi and only the second is 
celebrated. 



Corrections to Moon 307 

(1) Pratipada (Tithi 1) 

Caitra sukla pratipada i.e. that which precedes 
the Mesa Sankranti, is the beginning of Hindu 
Lunar year. New year's day (Lunar) falls on the 
day when pratipada is current on sunrise. When 
there is an adhika caitra, that begins the year. This 
tithi is, therefore, called Vatsararambha. It is also 
Navaratrarambha. 

There is another navavatra starting on Asvina 
sukla pratipada. 

Karttika sukla 1 is Balipratipada or Balipuja 
and is purva viddha as to time. 

Bhadrapada bahula 1 is Mahalayarambha. 
Phalguna bahula 1 is Vasantotsava 

(2) Dvitiya (Tithi 2) 

Asadha sukla 2 is Rathayatra dvitiya or Rama 
rathotsava. Kartika sukla 2 is yama dvitiya or Bhratr 
dvitiya (sisters make presents to brothers in 
afternoon) Bahula dvitiya in 'Asadha, Sravana, 
Bhadrapada and Asvina is called Asunya sayana- 
vrata and fast is broken at moon rise. 

(3) Tritiya (Tithi III) 

Caitra sukla 3 is gauritritiya, also Matsya 
jayantl (afternoon), also Manvadi (forenoon). 

Vaisakha sukla 3 is kalpadi (forenoon), Treta 
yugadi (forenoon), Aksaya tritiya (special when 
combined with Wednesday and Rohini naksatra, 
forenoon), also Parasurama jayanti. 

Jyestha sukla 3 is Rambha tritiya, when 
BhavanI is worshipped at purva viddha. 

Sravana sukla 3 is madhu srava in Gujrat. 



■4 



308 Siddhanta Darpana 



>?& 



-..-■s 



Sravana bahula 3 is kajjali tritlya 
Bhadrapada sukla 3 is varaha-jayanti (after- 
noon); Haritalika, when Parvatf is worshipped, 
Manvadi (forenoon). It is also called siva tithi. 
Phalguna bahula 3 is kalpadi (forenoon) 

(4) Caturthi (Tithi 4) 

Sukla Caturthi in every month is called 
Ganesh caturthi on Vinayaka caturthi, the chief 
being Magna Caturthi (Ganesa jayantl). It is 
celebrated at midday. Tila caturthi is its another 
name; but is observed in evening. It i* also called 
kunda raturthi. 

Bhadrapada sukla caturthi is special when it 
falls on Sunday or tuesday. 

Similarly, bahula caturthi in every month is 
Sankasta caturthi and is a fast day for people in 
difficulties. Fast is broken at moon rise. If it falls 
on tuesday, it called Angaraka caturthi and 
continues till moon rise. 

Sravana bahula caturthi is the main Bahula 
caturthi, and cows are worshipped. 

(5) Pancami (Tithi 51 

Caitra sukla 5 is Sri pancami. According t 
some, it is also kalpadi. 

Sravana sukla 5 is Naga pancami, when snak 
are worshipped. If the tithi starts within 6 gha 
after sunrise of one day and ends within 6 gh#! 
of sunrise on next day, the tithi is observed <* 
the first day. 

Bhadrapada sukla 5 is Rsi pancami. 



308 Siddhanta Darpatu^ 

Sravana bahula 3 is kajjall tritlya 

Bhadrapada sukla 3 is varaha-jayanti (after-? 
noon); Haritalika, when Parvati is worshipped, 
Manvadi (forenoon). It is also called siva tithi. 

Phalguna bahula 3 is kalpadi (forenoon) 

(4) Caturthi (Tithi 4) 

Sukla Caturthi in every month is called 
Ganesh caturthi on Vinayaka caturthi, the chief 
being Magna Caturthi (Ganesa jayanti). It is 
celebrated at midday. Tila caturthi is its another 
name; but is observed in evening. It i« also called 
kunda taturthi. 

Bhadrapada sukla caturthi is special when it 
falls on sunday or tuesday. 

Similarly, bahula caturthi in every month is 
Sankasta caturthi and is a fast day for people in 
difficulties. Fast is broken at moon rise. If it falls 
on tuesday, it called Angaraka caturthi and 
continues till moon rise. 

Sravana bahula caturthi is the main Bahula 
caturthi, and cows are worshipped. 

(5) Pancami (Tithi 51 

Caitra sukla 5 is Sri pancami. According to 
some, it is also kalpadi. 

Sravana sukla 5 is Naga pancami, when snakes 
are worshipped. If the tithi starts within 6 ghati 
after sunrise of one day and ends within 6 ghati 
of sunrise on next day, the tithi is observed, on 
the first day. 

Bhadrapada sukla 5 is Rsi pancami. 



320 Siddhanta Darpan 

bhadra. A sukla saptami on a sankranti is calle< 
Mahajaya which is superior to eclipse for makirtj 
donations. 

Vaisakha sukla 7 is Ganga saptami or Gangot- 
patti (birih of Ganga - midday). 

Sravana bahula 7 is Sitala or Sitala vrata, tiirM 
purva viddha. 

Bhadrapada sukla 7 is called Aparajita 

Asvina sukla 7 - About this tithi Sarasvati isl 
worshipped under mula naksatra 

Karttika sukla 7 is kalpadi (forenoon) 

Margasira sukla is Surya vrata. 

Magna sukla 7 is Ratha saptami or Ma] 
saptami (time* arunodaya), Manvadi (forenoon) 

(8) Astami (Tithi 8) 

An astami, falling on Wednesday, is specia| 
and receives the name of Budhastami. The Sukla- ' 
astami in every month is sacred to Durga or Am 
piirna, Bahula-Astami in every month calle< 
Krsnastami, celebrated at purvaviddha, is sacred t< 

• • - * * 

Krsna. 

• • • 

Caitra sukla 8 in 'Bhavani utpatti'; whe] 
joined with Wednesday and punarvasu naksatra, 
bathing on this tithi is special. 

Sravana bahula 8 - JanmastamI, Krsnastami cd 
Krsna Jayanti (midnight) special when combine^ 
with Rohini naksatra; less so when joined or^ 
monday or Wednesday. Manvadi (afternoon). i 

Bhadrapada sukla 8 - Jyestha Gauri pujana 
vrata; when combined with Jyestha naksatra. 



Corrections to Moon 321 

Bhadrapada bahula 8 - MahalaksmI vrata 
(purva viddha); Astaka sraddha. 

Asvina sukla 8 - Mahastaml, special when 
joined to tuesday. 

Karttika sukla 8 - Gopastami - worship of 
cows. 

Karttika bahula 8 - Krsnastaml, Kala 
bhairavastami or kala bhairava Jayanti. 

Margasira bahula 8 in Astaka sraddha in 
afternoon, the same is case with bahula 8 in Pausa, 
Magna or Phalgana. 

Pausa sukla 8 in special when on Wednesday 
with bharanl naksatra (Rohini or Ardra according 
to some). 

Magna sukla 8 is Bhlsmastami at midday. 

Magna bahula 8 is birth of Sita. 

(9) Navami (Tithi 9) 

Bhadrapada sukla 9 - Adukha navami. 

Asvina sukla 9 - Maha navami or Durga 
navami, Manvadi (forenoon) 

Karttika sukla 9 - Treta yugadi (forenoon) 
Margasirsa sukla 9 - Kalpadi (forenoon) 
Magha bahula 9 Ramadasa navami 

(10) Dasami (Tithi 10) 

Jyestha sukla 10 - Dasa-hara (destruction o\ 
10 sins) Ganga- vatara. 

Asadha sukla 10 - Manvadi (forenoon) 

Asvina sukla 10 - Vijayadasami (afternoon 
special with sravana naksatra, Buddha Jayanti. 



312 



Siddhanta Darpanm 



(11) Ekadasi (Tithi 11) 

Every Ekadasi is sacred and has a separate 
name. It is called Vijaya when combined with 
Punarvasu naksatra. 



-.(. 





Month 


Sukla 


Bahula 


1. 


Caitra 


Kamadi 


Varuthini 


2. 


Vaisakha 


Mohiiu 


Apara 


3. 


Jyestha 


Nirjala 


Yogini 


4. 


Asadha 

• • 


Visnu sayanotsava 


Kamadi or 






* 

Sayani or Visnu Sayani 


Kamika 






(Visnu going to sleep) 


- 


5. 


Sravana 

• 


Putradi 


Aja 


6. 


Bhadrapada 


Visnu parivartanotsava 
or parivartini (Visnu 


Indira 






turning on his side) 


* 




- 


called Visnu Srhkhala 

• • • 








when 11th and 12th tithis 








meet in Sravana naksatra 

• 




7. 


Asvina 


Papankusa or pasankusa 


Rama 


8. 


Karttika 


Prabodhini (Awakening of 
Visnu), Bhisma pancaka 
Vrata commences 


Utpatti 


9. 


Margasira 


Moksada 


Saphala 


10. 


Pausa 

• 




Putrada or Vaikuntha 

* • 


sat-tila 

• ■ 






ekadasi, Manvadi (forenoon) 


11. 


Magha 


Jaya 


Vijaya 


12. 


Phalguna 


Amalki 


Papa mocini 



12. Dvadasi (Tithi 12) 

This is called Mahadvadasi in the following 
circumstances - 

11th tithi current at sunrise on two . successive 
days : the next dvadasi is called Unmflani. 



Corrections to Moon 313 

12th tithi current at sunrise on two successive 
days - first dvadasi is called Vanjull. 

12th tithi followed by a full moon or a new 
moon tithi, current at two sunrises - Paksa 
Vardhini. 

12th tithi with Pusya naksatra - Jaya 

- do - Sravana naksatra- Vijaya 

- do - Punarvasu naksatra - Jayanti 

- do - RohinI naksastra - PapanasinI 

~ Vaisakha sukla 12, with Hasta naksastra, guru 
and mangala in simha, surya in mesa - Vyatlpata 

Asadha sukla 12, commencement of 
caturmasya vrata 

Sravana sukla 12, Visnoh pavitraropanam 

Bhadrapada sukla 12 - Vamana Jayanti (mid 
day); called Sravana divadasi when with sravana 
naksatra, specially on Wednesday. 

Asvina bahula 12 - Govatsa dvadasi (evening) 

Karttika sukla 12 - (i) End of caturmasya vrata 
which began on same tithi in Asadha. 

(ii) Prabodhotsava or Utthana dvadasi 
(preparation for waking Visnu) 

(iii) TulasI vivaha (Marriage of Visnu with 
TulasI plant) 

(iv) Manvadi (forenoon) 

Magna sukla 12 - Bhlsma dvadasi 

Magna bahula 12 - Tila dvadasi or vijaya when 
with Sravana naksatra (when previous Magna is 
adhika). 



314 Siddhanta Darpanm 

13. Trayodasi (Tithi 13) 

Caitra sukla 13 - Madana trayodasi or Ananga 
pujana Vrata (purva viddha) 

Bhadrapada bahula 13 - Kali yugadi (after- 
noon) 

(ii) Magha trayodasi - when with magna 
naksatra 

(iii) Gaja chaya .- when with magha naksatra 
and sun in hasta. 

Asvina bahula 13 - Dhana trayodasi 

Magha sukla 13 - Kalpadi (forenoon) 

Phalguna bahula 13 (i) Varum when joined 
with Satabhisaj 

(ii) Mahavarum - do - + Saturday 

(iii) Maha-maha -varum - when joined with 
Satabhisaja naksatra + Saturday + subha yoga 

(14) Caturdasi (Tithi 14) 

Bahula Caturdasi in every month is Sivaratri 

Vaisakha sukla 14 - Narasirhha Jayanti (sunset) 
: special when joined with svati naksatra + Saturday 

Sravana sukla 14 - Varahalaksmi Vrata 

Bhadrapada sukla 14 - Ananta caturdasi 3 

Asvina bahula 14 - Naraka caturdasi (moon 
rise), Dipavali may fall on this tithi if with svatf 
naksatra (normally on Asvina bahula 15) 

Karttika sukla 14 - Vaikuntha caturdasi (mid 
night) 

Margasirsa sukla 14 - Pasana Caturdasi 

Magha bahula 14 - Maha Sivaratri (mid night 
when Sravana naksatra is current). Special when 
combined with Sunday or tuesday and siva yoga. 



315 
Corrections to Moon 

15. Sukla PancadasI (Tithi 15) or Purnima 

A sukla 15 or purnima is called somavati when 
it falls on monday and is special for donations 

It is called cudamani - when further joined 
with a lunar eclipse. Special names are given below- 

Caitra Purnima 

(1) Manvadi (forenoon) 

(2) Hanumana Jayanti 

(3) Special for bathing when combined with 
sunday, thursday or Saturday. 

Vaisakha purnima - Kurma Jayanti (late after 

noon) 

Jyestha purnima (i) Manvadi (fore noon) 
(ii) Vata purnima or Vata savitri (purva 

viddha) 

(iii) Maha Jyestha when moon and jupiter are 

in Jyestha naksatra and sun in Rohinl 

Asadha Purnima (i) Manvadi (forenoon) 

(ii) Siva sayanotsava or Kokila vrata or Vyasa 

puja 

Sravana purnima (i) Rk yajuh Sravani (for 
followers of Rk and yajurveda 

(ii) Raksa bandhana (tying a string round the 
arm) or Rakhi purnima or narali purnima (throwing 
coconuts into the sea) 

(iii) Hayagrlva Jayanti 

Bhadrapada purnima - (i) Kojagari purnima 
or kojagara vrata (mid night) Laxmi and Indra 
worshipped; games of chance. 

(ii) Navanna purnima - when new grain is 
cooked. 



326 Siddhanta Darpana 

Karttika purnima (i) Manvadi (forenoon) 

(ii) Caturmasya vrata ends 

(iii) Tripurl purnima or tripurotsava 

(iv) Special when joined with Krttika naksatra 

(v) Maha Karttiki, when joined to naksatra 
Rohini or when moon and jupiter both are in 
Krttika naksatra. 

(vi) Padmaka yoga when moon in Krttika and 
sun in visakha 

Margasirsa purnima (i) Dattatreya or Datta 
Jayanti (evening) 

(ii) Special for donation of salt when joined 
with Mrgasira 

Magna purnima - maghi - when moon and 
jupiter both are in Magna naksatra 

Phalguna purnima (i) Manvadi (forenoon) 
(ii) Holika or Hutasani purnima (evening) 

(16) Bahula Pancadasi (Tithi 15) - Amavasya 

A solar eclipse on Sunday is cudamani and is 
special for donations. 

Sravana amavasya (at beginning of next month 
bhadrapada as in all cases) - Pithori or Kusotpatini 

Bhadrapada amavasya - Sarvapitri or Mahalaya 
amavasya, special when sun and moon both are 
in hasta 

Asvina amavasya - DTpavali, with previous or 
following tithis; that on svati naksatra is special. 

Pausa amavasya - (i) Ardhodaya when joined 
with Sunday in day time + Sravana naksatra + 
Vyatipata yoga. 

(This can happen only when some previous 
month is adhika) 



Corrections to Moon 327 

(ii) Mahodaya - when any one of these special 
features is lacking. 

Magna amavasya (i) Dvapara yugadi (after- 
noon) 

(ii) special for sraddha when joined with 
Satabhisaja or dhanistha naksatra. 

Phalguna amavasya - Manvadi (afternoon) 

Notes - (i) Many festivals differ due to 
interpratation by different sects or regions. 

(ii) The list is not exhaustive 

(iii) There were 14 manus (7 yet to come). So 
14 days are manvadi. 

(iv) Birthdays also differ according to inter- 
pretation whether original reckoning was solar or 
lunar. Birthdays of other sains like Tulasidasa, 
Nanaka, Kabira or Raidasa etc are also calebrated. 

(v) There are many other local festivals. 

C. Methods for Citation : (i) Normally the 
gata or expired years, amanta months and Caitradi 
years are given. A lunar year begins only when 
the solar year begins with mesa sankranti. Thus it 
is same as counting mesadi solar years. Compared 
to this, the current years in christian era are counted. 

(ii) Sometimes Varttamana or current years, 
purnimanta months or karttikadi lunar years are 
also given, peculiar to a system of samvat. 

(iii) An era is called year, samvat, samvatsara, 
san (arabic or persian), saka etc. Thus saka is not 
only saka era but year in any era is called saka. 

(iv) Samvatsara is named in 60 year cycle of 
guru varsa or Jovian years. In south India it is 
merely a solar year with Jovian name. In north 



318 Siddhanta Darpm 

India it is the Jovian year actually completed a| 
the beginnig of a solar year or year at the moment. 

Jovian years are also named in 12 years cycle, 
when it completes one revolution. The current rasi 
of Jupiter is name of year. Alternatively, Jupiter! 
years are named on lunar months, corresponding 
to solar rasis in which Jupiter is present. Maha is 
abided before these lunar months to indicate that 
they are years (Jovian) 

References : (i) Spherical astronomy by W.M. 
Smart, Longman Green, London or by R.V. Vaidya, 
Payal prakasana, Nagpur etc can be referred. 
Godfray has written a book 'Moon' only about 
Kinematic theory and perturbations of moon. 

(2) For finding correct positions, a Nautical 
Almanc can be referred. 

(3) History of calenders can be referred to the 
relevant article in Encyclopadia Britanica by 
Fotherington. History of calendar has also been 
published by govt, of India, publications division, 
being part C of report by National commission on 
calendars under Dr. M.N. Saha. 

(4) History of Astronomy by S.B. Diksita, 
Govt, of India, or by S.N. Sen and A.V. Sub- 
barayappa, published by Indian National Science 
Academy, Delhi - 2. 

(5) Bharata Varsa Ka Brhat Itihasa by 
Bhagavaddatta, Pranava Prakasana, Delhi - 26. 

(6) Indian chronology by L.D. Swami Kannu 
Pillai. 

(7) Reference for deciding festivals is Nirnaya 
Sindhu by Kamalakara Bhatta. 



Corrections to Moon 31 $ 

Translation of Text (Chapter 6) 

Verses 1-3 - Scope - Now I (author) write 
accurate panjika for getting quick results in 
marriage, sacred thread ceremony, house construc- 
tion, yajna and birth caremonies etc. With help of 
this, accurate position of sun and moon are known 
and kranti, sara, lunar and solar eclipse, conjunc- 
tion of planet and naksatra, rising and setting, 
mahapata, tithi, naksatra, yoga and karana etc can 
be calculated. 

Old astronomers assumed maximum increase 
of 5 danda and decrease of 6 danda in a normal 
tithi of 6 danda. Due to this their panjika was 
inaccurate, because actual increase or decrease limit 
of a tithi is much more. After describing rough 
panjika according to old school in last chapter, 
now I an explaining the method for accurate 
panjika. Calculation as per these rules will give 
correct time for auspicious works and the tithis etc 
can be actually seen. When direct observation 
proves the accuracy, no further logic is necessary 
in support of these rules. 

Mean sun and moon corrected only by 
mandaphala give correct position at amavasya and 
purnima. This has been described in previous 
chapter. 

Verses 4-6 - Need for further corrections 

Panjika has five limbs - vara, tithi, naksatra, 
yoga and karana - so it is called pancanga. Except 
the first part vara, all others depends on sun and 
moon. Hence these will be accurate if sun and 
moon are accurate. Manda paridhi of moon and 
sun is taken same for rough and accurate methods 



320 Siddhanta Darpana 

both. Hence sun and moon corrected by man- 
daphala is called 1st (corrected) planet. 

From this 1st graha and 1st ravi sphuta 
gati, we calculate diameter of planet, time upto 
parva sandhi (i.e. purnima or amavasya), bhuja, 
mandakarna and lambana correction of sun. 

* 

Motion of moon is very complicated. After 
long period of observation, I have thought it 
necessary to have 3 more correction in addition to 
mandaphala. These four corrections are - Manda, 
Tungantara, Paksika and digamsa. 

Verses 7-9 - Tungantara correction 

Tungantara kendra = Candra Mandocca 

- (sphuta ravi + 3 rasi) - in Suklapaksa 

or = Candra mandocca - (sphuta ravi - 3 rasi)! 

- in Krsna paksa 

Find out bhuja jya of tungantara kendra. j 

Tungantara bhujaphala (its bhuja jya) isj 
multiplied by 16 and divided by radius (3438). Then| 
it is multiplied by bhuja jya of difference of sphutaf 
ravi and sphuta candra and again divided byj 
radius. 1 

"4 

Result in kala etc. is multiplied by 1st candraj 
sphuta gati and divided by madhya candra gati| 
(790'i35"). Result in kala etc will be tungantara| 
phala. When tungantara kendra is 0° to 180°, thte| 
is added to 1st sphuta candra otherwise substracted.| 
We get second sphuta candra. 1 

Verses 10-12 - Paksika phala J 

Paksika Kendra = 2nd sphuta candra - sphut^ 
ravi. Lapsed and remaining parts of the kendra 
its quadrant are found. Lesser of the two 




Corrections to Moon 321 

converted to kala and multiplied by 2. Jya of the 
resulting angle is divided by the hara, to be 
calculated as below, gives paksika phala in kala 
etc. When candra (2nd) is in 1st half of paksa, 
paksika phala is added to 2nd candra, otherwise 
it is deducted. 

Paksika hara is found by substracting 1st 
sphuta sun separately from mandocca of surya and 
candra. Bhujajya of the two remainders is calculated 
and multiplied together. Product is divided by 180 
and to the quotient we add 90. Result is the hara 
for paksika phala. 

Verse 13 Digamsa phala * 

Mandaphala calculated from sphuta sun is 
divided by 10; multiplied by sphuta candragati, 
and divided by madhya candra gati. Result is 
digamsa phala. The mandaphala being positive or 
negative, digamsa phala is added or substracted 
from 3rd sphuta candra. We get 4th sphuta candra 
which will be accurate position. 

Verses 14-15 : Reasons for correction - 

In plane surface snake moves in a wave like 
motion, but at the time of entering a hole, its 
motion becomes straight. Similarly, moon deviates 
from the mandocca resultant motion normally, but 
on purnima and amavasya, these deviations vanish 
and only mandocca effect remains. 

When snake enters a hole, its wavy motion 
ceases under pressure from narrow sides, but its 
natural forward motion also is affected. Similarly 
on parvasandhi, moon is not affected by tungantara 
and paksika sanskara, but digamsa phala is still 
effective (in addition to mandaphala). 



% 



322 Siddhdnta Darparui 

Comments : Fortnightly venations in orbit due 
to sun effect are not evident on parvasariclhi 
(purnima or amavasya) because sun, moon and 
earth are in a straight line. Howevere, total 
attraction of sun, varies according to its own 
variation in distance, causing minor corrections of 
digamsa phala. Parvasandhis are like a hole for 
snake, hence the simili. 

Verse 16 - Rough and accurate correction for 

sun - For correcting accurate sphuta of ravi, 
accurate chart should be used and for rough 
sphuta, rough chart is used. Use in reverse order 
will create errors. 

Verses 17-21 - Accurate sphuta gati of candra - 

.Accurate tungantara phala is multiplied by 
radius (3438) and divided by Jya of difference 
between 1st sphuta candra and accurate sphuta 
ravi. Result is multiplied by kotijya of difference 
of (1st sphuta ravi - candra) and divided by radius 
(3438). Result added to 1st sphuta candra gati phala 
gives 2nd gati phala. This is added when rnanda 
kendra is 90° to 270°, otherwise substr acted. 

Paksika phala in kala etc. is squared arid 
deducted from the square of maximum piksifca 
phala. Square root of remainder is multiplied by 
difference of 2nd candragati and sphuta sufya gala 
and divided by half radius (1719). Result is adde«f 
to 2nd gati when candra is in 1st half of Sukia 
paksa or 2nd half of krsna paksa. Otherwise, it is 
deducted. This is third sphuta gati of moon. This 
*third gati will be accurate. Difference of sphuta 
candra on two successive days also gives suffiderit- 



Corrections to Moon 323 

ly accurate gati for calculation of tithi and auspi- 
cious works. 

Comments : Corrections have already been 
explained in the introduction of this chapter. This 
explains the change of speed due to two paksika 
variations due to sun's attraction. Last correction 
is due to change in distance of sun which is 
negligible within a day and correction is not needed 
for r already small effect. 

Tungantara phala = - 160' cos (0—a) Sin (D-0) 

where D = moon corrected for mandaphala, 
6- , Longitude of sun, 

a = mandocca of moon. 

For a short period, only D is variable which 
is position of moon. 

Hence speed due to this correction is by 
differentieting Sin (D-0) 

= - 160' cos (0-a) Cos (D-<9), d(D~0) 

RxTungantara phala R cos (D - 0) 

= Rsin(D-0) ' R 

Thus we get the formula 

Paksika phala = 38'12" Sin 2 (D-0), 38'12" = 
max phala = P 

where D' = 2nd sphuta moon. 

Gati due to paksika phala is its differntial 

= 38'12" cos 2 (D-6) x 2 d(D-0) 

= 38'12" / 1 _ sin 2 2 (D _0 } x 2 (dD - dS) 



= J Vp 2 - (Paksika phala) 2 x ( 2n d candra 
gati - ravigati) x 2 






324 Siddhanta Darpana 

r —— 2nd candra gati - Ravi gati 

= V P 2 - Paksika Phala) 2 x wi 

This is the formula given above 

Verse 22 : Sthula value is not entirely useless, 
it is good for daily use. But adverse moments like { 
visti, to be strictly avoided, should be calculated j 
only through accurate motion. 

Verse 23 : Phases of moon 

Sun deducted from moon gives the kendra, 
when this kendra is in ist 6 rasi i.e. 0° to 180°, it 
is sukla paksa. When moon is ahead of ravi by 
180° to 360° it is krsna paksa.First 3 rasis are 1st 
half of suklapaksa, then upto 6 rasis it is 2nd half 
of sukla paksa. Similarly in krsna paksa, Ist half 
is from 6 to 9 rasis and 2nd half is from 9 to 12 
rasis. One fourth of every paksa (i.e. 45° difference) 
is called paksa pada (quarter). 

Verses 24-26 - More correct motion of moon - 

Now I am telling more accurate motion of 
moon which needs to be calculated for eclipse. For 
this gati, we find gata and gamya kala (lapsed and 
remaining) times of parvanta (purnima or 
amavasya). From that, true moon is found out. 
From this sphuta gati, sparsa, moksa, sthiti etc 
periods of eclipse are accurately known. 

First gatiphala is kept in two places. At one 
place it is multiplied by parama tungantara phala 
(160) and divided by parama mandaphala (300'50")/ 
At second place it is multiplied by Ist sphuta gati 
and divided by madhyagati (790'35"). Result of 
both places are added. The sum is added to 
madhyama gati of candra when manda kendra is 



Corrections to Moon 325 

in 6 rasis starting from karka (90° -270°), otherwise 
it is deducted. This is true mean speed of moon. 

This is again kept at two places. At one place, 
sphuta surya gati is substracted and divided by 
half of hara. Here Hara = bhuja of (candra 
mandocca - suksma surya) x Bhuja of (ravi man- 
docca - suksma ravi ^180° +90°. Quotient is added 
to the true mean motion of moon at second place. 
Sum is true motion of moon. 

Kotiphala x Kendragati 
Comments - 1st gatiphala = — : — 

r .cos m x 6m 

= _ 

Thus the above formula is 

<5 m 160 r cos m 

— — - r cos m ( + 1 + — - ) 

R v r R. 

Gatiphala of Tungantara is 

160 cos m. cos (D-0) 

m = D - a = mandakendra, 

cos (D-#) is 1 for parvanta as D-^ = 0° or 
180° 

d M 
Hence, tungantara gati phala = 160 cos — r— 

This is the first term of above formula • 

d M 160 160 cos m s 

— — - . r cos m . = — dm 

R r R 

Remaining terms are second order corrections 
in mandaphala itself. 

Second step of correction amounts to paksika 
correction as stated earlier after verse 21. 



326 Siddhanta Darpana 

Verse 27 - Accurate panjika - Use of one 

rough panjika for normal works and another 
accurate panjika for important works will not be 
appreciated by anybody. Hence only accurate 
panjika should be used, even though it involves 
hard labour. It alway deserves more respect. 

Verse 28 - Definition' of true planet * We are 

on the surface of earth. The point of sky where 
line from earth's interior centre to our location 
point on surface meets is called 'svastika' (vertically 
upward point). When the planet is seen on the 
great circle from kadamba (pole of ecliptic) through 
svastika and calculated position of graha on ecliptic, 
graha is called spasta. 

Note - This will be explained fully in chapter 
7 and lambana samskara for solar eclipse. (Chapter 

9) 

Verse 29-31 : Need for calculating true planet - 

According to old teachers, all auspicious works 
are done only according to this true graha. For 
this purpose bhagana and bija corrections are done 
to the planet. 

For daily and special works of vaidika and 
smartta type, true position of all planets are 
needed. But correction to moon is needed more, 
because pancanga is based on moon's position. 
Hence, accurate corrections like tungantara, paksika 
and digamsa etc. have been thought of. 

Even after these three sanskaras, there is 
difference of 2-3 palas (upto 1 minute) in the 
calculated and observed position of moon. But it 
is preferable to error of upto 14 ghati (about 6 
hours) which will occur without these corrections. 



327 

Corrections to Moon 0£ -' 

Only Brahma can know how to eliminate this small 

error. 

Note : Though these corrections are great 
improvement, some error will always remain. Error 
withhin 1 minute is sufficient for day to clay work. 
More accurate position is needed for scientific 
works. Every formula will give some error, though 
it is about 1/10 seconds or less in modern methods. 

Verse 32-33 : Lambana and sara 

A planet will be seen in different position, 
when seen from earth's centre (which is calculated) 
and when seen from surface (where we are 
located). 

The angular difference between two position 
is called lambana. (In vertical position, it is already 
in line from earth centre to surface, hence lambana 
will be nil). 

Distance of the planet from kranti vrtta along 
great circle from pole of ecliptic (kadamba) to centre 
of planetary disc (also passing through svastika - 
vertical up point) is called sara or viksepa. 

Calculation of lambana and sara is called 
drk-karma (change of axis). This is needed only 
for lunar and solar eclipse and conjunction of 
planets. In that context only, it will be calculated. 
It is not needed for calculation of tithi and naksatra 
etc. 

Verses 34-39 : Authorities on need of true 
planets - Vrhatsamhita (Varahamihira) has stated - 
If grahana occurs before calculated time, then 
damage to foetus or child in womb, or war with 
weapons occurs. If it occurs after calculated time, 



32 ° Siddkanta Darpana 

then damage to crops, loss of flowers and fruits 
and fear for people occurs. 

Garga sarhhita states - The result of having 
eclipse before or after calculated time has already 
been stated. Persons knowing true planets, never 
have error in timing. If every (astronomical) event 
occurs according to calculated time, then enemies 
of kings are destroyed and troubles cease. People 
become happy, being free from fear and disease. 

Vasistha states (not known in which text) - 
Tithi etc. should be decided according to that 
theory only which gives true position of planets. 

Sakalya samhita states - Corrections to calcu- 
lated position of planets should be done after 
observing them through instruments like nalika 
(tube or telescope) gola (mirrors or sphere), turiya 
(Fourth - compound telescope). Correction of 
observed error is called bija sanskara. After that 
correction only, all rules will be correct. Result of 
direct observation (pratyaksa) cannot be ignored. 

What is use of the gold ornament which cuts 
the ears ? Similarly what is the use of that sastra 
whose results are not actually seen ? 

Verses 40-42 : No need of lambana for tithi cal- 
culation 

The people who talk of lambana for calculation 
of tithi etc, do not know its meaning. Explaining 
them is like talking to a deaf. 

Spasta graha is known from the point of 
intersection of line from earth centre to the planet, 
when the graha is seen at that place, it is called 
spasta (or true) planet. 



Corrections to Moon 329 

If graha is seen from earth's surface, tithi will 
be different for different places due to separate 
larrtbana corrections, hence tithi needs to be 
calculated from earth centre, so that it is same all 
ove the world. 

Verses 43-46 : Bija sanakara - 

When graha is not same as per ganita 
(calculation) and drk (observatoin), it cannot be 
used for auspicious works. So I describe the bija 
karma, i.e. corrections to calculated position to tally 
it with observed position. 

Bhaskracarya (in BIjopanayana) has stated - 
After daily observations of moon, I have observed 
that moon is seen 112' lipta east or west from its 
calculated position. These are the minimum or 
maximum values of Bija. 

In Surya siddhanta - Sun himself has stated 
in the end that he was explaining bija for good of 
the world even though it was a secret; after praying 
to gods and vedas. 

In Brahma-sphuta siddhanta - Graha ganita 
(calculation from planetary theory) as told by 
Brahma himself was lost (became erroneous) after 
lapse of long time. So Brahma gupta, son of Jisnu, 
seeks to correct it with bija-sanskara. 

Verse 47-52 : Origin of Tungantara correction - 

The error in calculated position of moon upto 
112' Hpta (stated by Bhaskara) is probably due to 
distance of moon from ecliptic, so it should be 
related to the maximum viksepa (281' lipta). 
Because, after adding 3 rasis of sayana moon, Jya 
°f its kranti is 1370'. This multiplied by maximum 



330 . Siddhanta Darpana 

viksepa and divided by radius (3438) gives 112 
kala. This is the same amount which is found by 
calculating difference between calculated planet and 
observed planet. 

Bhaskarcayra has called it Ayana dik—karma, 
there are different types of Ayana karma in other 
siddhantas. So it should be called a bija sanskara. 
I have called it parama tungantara phala. 

It appears from drk-karma of Bhaskaracarya 
that maximum value of tungantara sanskara is 
continuing since long ago. According to ancient 
teachers, it fluctuates, so they have advised to 
correct moon with bija sanskara. 

To know the change in maximum value of 
tungantara phala, moon will be corrected after one 
thousand years. The error from true moon will 
give the value of change. 

Notes : Candrasekhara has not understood 
the theory or reasons behind this tungantara 
correction, But from the nature of variations, he 
has correctly assumed the position of maximum 
deviation and hence has got the correct formula. 

Verses 53-57 - Variations in duration of tithi - 

A tithi which includes aparahna of two 
consecutive days has beeen called suksma tithi in 
smrtis. Thus such a tithi has more than 66 dandas 

* • ■ 

(as aparahna period is 6 danda and a day is of 60 
dandas). 

According to smrti, if tithi just touches one 
evening and is over before first half of day, then 
the sraddha of that day should be done on next 
day. It should be over by 'kutupa' of next day. 



Corrections to Moon 331 

(Here 'Kutupa' means 8th muhurtta of the day 
time out of 15 muhurtta = 30 dandas between 
sunrise to sunset). Thus tithi = sayahna 6 danda 
+ night 30 danda + half day time 15 danda = 51 
danda. 

Gautama smrti has stated - If in sayahna 
(evening) of caturdasi (Krsna paksa), amavasya 
starts and is over before midday, then sraddha 
should begin in kutupa muhurtta (14 to 16 dandas 
from sunrise) and should be over by rohana 
muhurtta (12-14 danda after sunrise) on next day. 
This is called amavasya sraddha. 

In sukla and krsna paksa, on 7th, 8th and 
9th - the three middle tithis, maximum difference 
in tithi duration is less than 6 danda. So these 
tithis have only 5 types of classifications - the 6th 
category of above six danda difference from 60 
danda doesn't exist. Other 12 tithis in both paksa 
have 6 types 6f class. If smrtis are interpreted in 
this manner, there is no error in drk siddha (true) 
calculations. 

Ancient teachers, didn't observe the daily 
location of moon in constellations. With rough 
calculation, there is variations in middle tithis also 
upto 14 dandas (i.e. I 7 danda from average). But 
they had strived for accuracy, only at the end of 
a paksa when wrong eclipse time will cause insult 
to the astronomer. 

Notes - Traditional view about variation of 
tithi is 'Bana vrddhi, rasaksayah' i.e. increase upto 
5 dandas and decrease upto 6 danda. This gives 
tithi limit from 54 to 65 dandas. But Candrasekhara 



332 Siddhdnta Darpana 

has found corroboration from smrtis that it is 
actually from 51 to over 66* dandas. 

* 

Verses 58-67 : Suksma naksatra of unequal 
divisions - Now I explain the method to calculate 
suksma naksatra (unqeual divisions) for use in 
journey, marriage and sacred thread ceremoney etc 
as decided by sages like Garga, Vasistha. 

Mean motion of moon in a day (790'35") is 
the extent of suksma naksatra. One and half times 
this value is the extent of these six naksatras equal 
to (1185'52"18"') - (4) RohinI (7) Punarvasu (16) 
Anuradha and three Uttara naksatras (12) Uttara 
phalgunl (21) Uttarasadha (26) Uttara Bhadrapada. 

Half extent (395'17"26"') is of the six naksatras 
(9) Asiesa (15) Svati (18) J^estha and (24) Satabhisa 

Remaining 15 naksatras have unit extent 
(790'35"). Deducting the total of these 27 naksatras 
(21345'41"5'") from kalas of full circle (21,600), 
remainder (254'18"35 ,,/ ) is the extent of Abhijita 
which comes between (21) Uttarasadha and (22) 
Sravana. 

We substract the kala of as many naksatras 
from sphuta graha as it is possible. It is the number 
of completed naksatras. Remainder (gata) kala of 
the graha is the lapsed part of current naksatra. 
This part deducted from full extent of current 
naksatra gives remaining part (gamy a or bhogya 
kala). Gata and bhogya kala, separately multiplied 
by 60 and divided by sphuta gati of graha, give 
the lapsed or remaining time of the graha in current 
naksatra. 

Each of the 28 naksatras of unit, half, one 
half length or Abhijit being divided by 4 gives its 
one pada (quarter). 



<>i 



Corrections to Moon 

As per rough rule, rasi of 1800 kala contains 
9 naksatra pada (27 naksatras excluding Abhijit 
have 27 X 4 = 108 pada = 12 rasi X 9). Thus 108 
pada in 12 rasis are according to mean equal values 
of naksatras. With suksma rule, 1 rasi doesn't have 
complete number of naksatra padas. 

For example, at the end of 4 rasis 4X9 = 
36 naksatra pada or 36 -s- 4 = 9 naksatra till Aslesa 
will be completed and 5th rasi sirhha should start 
with 10th naksatra. But according to suksma 
calculation magna naksatra starts 8° before simha 
rasi itself. 

Notes : (1) There are three measures of a 
pancanga for approximately one civil day. Week 
days are for fixing current routine of work and a 
day more than seven days ago or in future is not 
referred by the week day. Thus in modern 
university history books, even for modern eras, 
week days do not figure. This doesn't mean that 
use of week days is not common in modern days. 
Due to temporary nature of weekdays, and use for 
astrology only, they have not mentioned in 
histories of Ramayana and Mahabharata and in 
vedas. This should not mean that week days were 
not known in ancient India, as it is concluded by 
so called modern scholars. Another weakness of a 
week day is that it starts from local sunrise time 
in Indian system (local midnight in Gregorian or 
christian calender, local evening in Jewish and 
Islamic calender). Thus in all systems, it starts at 
different time at different places. Thus it cannot 
be made a world rference. 

(2) Technically, tithi starts on same time all 
over the world. But for civil purposes, only the 
tithi current at sunrise is counted, hence it may 



SS4 SiddhdntQ Darpatyi 

■ i 

be useful for religious functions, but civil tithi willj 
be different in different places. Another defect is! 
that it is a mathematical calculation, even when! 
moon has risen, only the approximate tithi can be i 
known from its phase by rough eye estimate. 
However, naksatra can be measured more accurate- 
ly even with seeing moon's position among stars; 
by naked eye. Even for calculation purpose, it! 
doesn't suffer the errors in finding true position^ 
of moon, as it can be seen by direct observation,- 
This is the reason that all the important events in 
Mahabharata, Ramayana and Pur ana are indicated 
by naksatra of moon (instead of tithi) in addition 
of the lunar month. This is evident to the whole 
public and easily identifiable time in distant future.; 

(3) When naksatra extent is made exactly equalf 
to the mean motion of moon in one day, some 
part of the full circle will be left out as the moonj 
takes more than 27 days for a siderial revolitions. \ 
Thus 27 naksatra equivalent to 27 days motion of 
moon, doesn't cover the circle completely and a 
small 28th naksatra abhijit is introduced equivalent 
to extra time beyond 27 days taken in moon's 
sidereal revolution. 

Reason of unequal division is that, along the 
path of moon in sky, inclined at 5° angle with 
path of ecliptic, sufficient bright stars are notj 
available for all naksatra divisions. The three vacant^ 
places were identified with their preceding stari 
groups causing division of 3 naksatras in purva^ 
and uttara parts - Phalguni, asadha and bhadrapada 
(or prostha pada - old name). These three vacant 
places and 3 other star groups having lesser gaps 
- were given 1-1/2 times the length. Correspond- 



1 



Corrections to Moon 335 

ingly, the length of 6 naksatras was reduce to half 
to compenisate the excess. 

(4) Visvamitra made equal divisions for each 
of 27 naksatras and further divided them into 124 
parts each for accurate calculations of solar and 
lunar naksatras at the end of day, paksa or half 
year. Thus he created a different naksatra system 
- which is proverbial creation of stars by him. This 
has been explained in introduction, while explain- 
ing vedanga jyotisa. Corresponding to 24 original 
naksatras, there are 24 letters in a Gayatri chanda. 
But with 3 extra naksatras by division of 3 into 
purva uttara parts, 3 extra letters (vyahrtis) were 
added to Gayatri mantra whose sage is Visvamitra, 
making 27 letters in it. This corroborates the view 
that number of verses in Rk veda and number of 
letter in its chandas are based on astronomical 
measurements at regular intervals. This unequal 
division will be more clear when longitudes of 
identifying stars are discussed. 

Verses 68 - 71 - Sankranti i.e. crossing from 1 
division to another 

Exact point of sankranti of a rasi is when 
centre point of a planet's disc reaches the last point 
of the rasi. This suksma sankramana is known by 
name of rasi which is to be entered, not the past 
rasi. Complete sankranti period is the time taken 
by complete disc of graha from touching the border 
point to its complete crossing. To find the 
sankramana kala, drameter of graha birhba (disc) 
in vikala is divided by graha gati in kala. 
Sankramana kala is obtained in danda etc. Within 



336 Siddhanta Darpar 




■!? 



sankranti period, surya gives very favourable! 
results. 

Planets give mixed results of both rasis during 
sahkranti period. Similarly while crossing over from 
one naksatra to the next, as long as the border 
point is covered by birhba (disc) of the planet, it 
gives results of both the naksatras. 

Candra birnba (disc) in Vikala, separately 
being divided by (1) difference of ravi and candra 
gati (2) candragati, (3) sum of candra and ravi gati, 
gives respectively the sandhi time of (i) tithi or 
karana (2) naksatra and (3) yoga. 

Verses 72-74 - Different circles - Due to effects 
of ucca, kranti and pata, many circular orbits are 
formed. 

Orbit due to attraction of slghra and manda 
ucca is called pratimandala (eccentric circle - 
explained in previous chapter). Path of kranti is 
called apamandala (to be explained in 
Triprasnadhikara).' 

Due to deviation of graha from kranti vrtta 
due to pata, another circle vimandala is formed 
which is path of pata (apamandala and vimandala 
are great circles perpendicular to ecliptic and will 
be explaiend in next chapter. 

Verses 75-91 : Precession of ecliptic and 
Ayanamsa 

Point of . intersection of kranti vrtta (ecliptic 
plane of sun's orbit) and visuva vrtta is. called pata 
which moves in the opposite direction to the 
normal motion of planets. Completed revolutions 
(bhagana) of pata in a kalpa are (6,40,170) as 
observed by the author. 



Corrections to Moon 337 

This pata is above all planets and circle of 
naksatras (slowest rotation indicates farthest dis- 
tance). This moves the naksatra vrtta from east to 
west in plane of kranti vrtta. 

When this pata is in six rasis beginning with 
mesa (0° to 180°), it takes the naksatra and planets 
etc 27° towards east. When it is in six rasis starting 
from tula (180° to 360°), it takes the naksatras etc 
27° towards west. 

Due to this pata, planets like ravi and 
naksatras starting with Asvini are seen towards 
north or south from ecliptic even on the position 
of 0° kranti. 

To find out kranti pata for desired day, kalpa 
revolutions of kranti pata are multiplied by 
ahargana and divided by savana dinas in a kalpa 
(i.e. 15,77, 91, 78, 28, 000). We get the complete 
revolution numbers and from remainder rasi etc. 
of kranti pata. 

The result in rasi etc is substracted from 12 
rasi and remainder is converted to bhuja according 
to quadrant and then to kala. Bhuja kali divided 
by 200 is called calansa. This is also called 
ayanamsa. 

Raminder after division by 200, is multiplied 
by 60 and divided by 200. We get kala of ayanamsa. 
Motion of ayanamsa in one day is 9/28 para etc. 
At the beginning of karanaba (1869 AD, mesa 
sankranti at Lanka), ayanamsa was 22°1'51"45'"42"" 
etc. 

When krantipata is in six rasis beginning with 
tula, then ayanamsa is negative and, when in six 
rasis beginning with mesa it is positive. 



338 



Siddhdnta Darpana 



According to Surya siddhanta - Ayanamsa 
corrected graha (or say ana graha) only is used for 
calculating kranti, chaya, carkhanda etc. Motion of 
kranti pata can be seen at the time of visuva 
sankranti (sayana karka sankranti in uttarayana and 
makara sankranti in daksina ayana). 

Saptarsi, Agastya and Yama and the stars 
close to them have no motion due to kranti pata 
(They are near north or south pole and very far 
from ecliptic). Their motion in naksatra mandala 
towards east indicates that naksatra circle has 
moved west wards. Seeing west ward motion of 
Saptarsi etc means that naksatra circle has moved 
eastwards. With this concept, astronomers calculate 
the sara of naksara, which is north or south 
deviation from kranti vrtta along circle perpan- 
dicular to it. 

Position of sun calculated from shadow 
(chaya) is different from mathematical position of 
true sun. This difference is ayanamsa. This 
ayanamsa is also moving eastwards. If calculated 
true sun is more than sun found from shadow, 
then ayanamsa is moving west wards. 

At the time of karka 
and makara sankranti, 
when kranti of sun is 
equal, the rasi etc of sun 
at both points is added, ei 
Their half is ayanamsa. E2 / 
When sayana karka or£ 
makara sankranti is seen 
before nirayana 

sankrantis then ayanamsa 
will be added, otherwise 
it will be deducted. Figure 2 




Corrections to Moon 339 

Note - (1) Newton's explaination : (Figure 2) 
C is pole of ecliptic EL'L. Let Ti be mid point of 
E and L and thus the first point of mesa for year 
1 . Then the celestial pole is at Pi and celestial 
equator is Ei,TiQi Due to precession of equinoxes, 
the first point of mesa is slowly moving in 
backward direction LT 2 E along the ecliptic. If Ti 
shifts to T 2 in year 2, the celestial pole shifts to 

P 2 along a small circle P1P2P3 where CP is 

obliquity of the ecliptic. The celestial equator 
assumes a new position E2 T2 Q2 in year 2. The 
celestial pole Pi goes round the pole of ecliptic C 
and it makes a complete circle in a period of about 
26000 years. 

R 




Figure 3 
In Fig 3 - if earth is homogenous sphere, the 
force of attraction of sun will act, as if the mass 
is concentrated on its centre C, But it is an oblate 
spheriod, whose polar axis is shorter than 
equatorial axis by 43 Kms. The main pull due to 
sun is still along CS which keeps earth in orbit 
round the sun. But the bulge at equator EEi suffers 
additional pull. The nearer portion of bulge at Ei 
1S attracted more and E less. This extra forces at 
Ei and E are equal and opposite in the direction 
°f sun, but line EEi is inclined at an angle with 
C S. Hence it is a couple which tries to bring earth's 



-■A'* 



340 Siddhdnta Darpana ■} 

equator in plane of ecliptic. Due to this couple, j 
precessional motion arises. 

Overall reason of precession of solar orbits is 
that each planet influences the other and net effect : 
is to bring angular momentum vector of all planets 
nearer to the direction of total angular momentum 
of the solar system. This mutual perturbation has 
a cycle of around 28,000 years. Due to motion of 
sun round the galactic centre also the angular 
momentum vector of solar system is turning in 
direction of galaxy's momentum. However this 
effect is very small and occurs in a period of about 
250 million years. 

Rigid Body Dynamics by A.G. Webster gives 
the following formula - Angle of precession PiCP 2 
= *¥ due to sun's attraction 

3ym C - A sin 21 v 

ip - — £— x cos (o (t - -r— ) 

2QT 3 C zn 

where = y gravitational constant = 6.67 X 10" 8 

C.G.S. units 

C = moment of inertia of earth round the 

polar axis 

A = moment of inertia of earth round an 

equatorial axis 

o) = Obliquity of ecliptic = 23°26'45" 
m = mass of sun = 1.99 X 10 33 gms 
r = distance of earth from sun = 1.49 X 10 



cms 



y m 



-—- = tide raising term 

1 = longitude of the sun 

n = angular velocity of earth in orbit 



Corrections to Moon 341 

Q = angular rotational speed of earth in 
radians 

For a homogenous sphere, C=A and *P = 0. 
If polar radius C = a (l-£), where e is ellipticity of 
earth, 

C - A 1 

= e = — if concentric layers of earth 

C 297 J 

are assumed homogenous. Bui its real value has 

been found to be — ~r. Putting the values in formula, 

304 ° 

dWs 



dt 



due to sun is 



^^ x ^— — cos © (1 - cos 21)= 2.46xl0' 12 rad/sec. 

Qr 3 C v ' 

It is multiplied by 2.063 X 10 5 = seconds in 
radian and 3.156 x 10 7 seconds in a year to get 
seconds of arc per year. Thus rate of solar 
precession = 16".0 per year. 



p m 



The tide raising force ^^ for moon is more 

than double of the sun. Thus lunar precession = 
34". 4 per year. 

Moon's orbit is making an angle of 5° 9' 
average with sun's path (ecliptic) varying ± 10'. 
Point of interaction of moon's orbit travels on 
eecliptic in a period of 18.6 years (motion of rahu). 
Figure 4 shows G, M as poles of ecliptic and of 
moon's orbit. ,P as celestial pole (earth north pole). 
Solar precession is vector along line PS. perpen- 
dicular to CP, lunar precession is represented by 
vector PR which goes up and down as M goes 
round C in a cycle of 18.. 6 years (Rahu period) 
components of motion are 



342 Siddhanta Darpana 

Along PS. = ¥ms = ^s + Wm CosM PC 

Perp to PS. *Pn = ^m Sin MPC 

This causes certain irregulaties in precessional 
motion and also in the annual variation of obliquity 
- which is called nutation - with a period of 18.5 

years 

If t = no. of years after 1900 AD, then 
Rate of precession = 50". 2564 + 0" .0002225 t 
Angle between equator and ecliptic planes is 
23°27'8".26 - 0".468 t 
Correction in precession due to nutation is 



Figure 4 

-17".235 sin (sayana rahu) - 1".27 sin (2 sayana 
sun) 

Correction in incline of equator is 

+ 9". 21 cos (sayana rahu) +' 0".55 Cos (2 
sayana sun) 

(2) Indian theories of precession : Correct 
theories : One theory states continuous backwarct 
motion which is correct as per modern theory. 
Other theory indicates -oscillatory motion when is 
not correct either according to modern theory nor 
accordiing to references in Vedas or brahmanas. • 



Corrections to Moon 343 

Rates of Steady precession ; Various quota- 
tions from pur anas, brahmanas indicate different 
position of equinoxes. 

Rgveda tells rains from Mrga naksatra (1-161- 
13). Taittiriya samhita (17-4-8) indicates vasanta at 
phalguna full moon. Both indicate a period of 
23,720 B.C. when equinox was 352° behind present 
position. 

Valmiki Ramayana indicates demon dynasty 
with Mula naksatra at vernal equinox. This should 
occur at 17000 B.C. which tallies with Egyptian 
countings mentioned by Herodotus. It also tells 
beginning of Dcsvaku dynasty with vernal eqninox 
at visakha at about 15080 B.C. This was the time 
of great deluge which is correct as per geology 
and sumerian records. 

Mahabharata indicates fall of pole star vega 
(Abhijit). At about 12,400 B.C. this was the pole 
star. Hence, around this star, a small extra naksatra 
had been assumed. 

Taittiriya Brahmana (1-5-2,6,) states krttika to 
visakha are Deva naksatras which turn Sun from 
south. Anuradha to Apabharani are yama naksatra 
which turn sun from north. This position of winter 
and summer solstice was in 8357 B.C. Varah- 
amihira tells that winter solistice was at Dhanistha 
beginning at time of Vedanga jyotisa and at Makara 
beginning in his own time (about 100 B.C.). He 
has concluded backward motion of ay ana. 

Satapatha Brahmana tells Krttika at equator, 
present position being 36° 9' east and viksepa 4° 2'. 
This was about 67° 56' east of present position of 
vernal equinox. This was 2942 B.C. 



344 Siddhanta Darpana 



J 



Manjula (932 AD) has indicated backward 
precession of vernal equinox 1,99,669 cycles in a 
kalpa i.e. 59". 86 per year. Bhaskara II has also 
accepted his authority. He has stated that ayana 
was non existant at time of Aryabhata and 
negligible at time of Brahmagupta and so they have 
not discussed. Even Bhaskara has mentioned it 
only in the context of constructing gola bandha 
(armillary sphere) Curiously Jagannatha Samrata in 
his Siddhanta Samrata has indicated 278 Saka as 
year of zero Ayanamsa and rate of precession per 
year as 51". This value is accepted as per modern 
calculations Prthudaka (928 A.D.) has given 56."82 
per year. 

Even Munjala value is very accurate. In 932 
A.D. yearly rate of precession was 50.2453- 
0.0002225 t (years from 1850 AD) = 50.041". 
According to Indian practice, excess precession for 
tropical year is 9.76", then correct precession should 
be 59.8" per year which is very close to his value 
of 59".86. 

Liberation theory : A suspect passage occurs 
in Surya siddhanta, Triprasnadhikara, (9-10) which 
states - 

In a yuga, naksatra cycle falls back eastward 
thirty scores (f^^TT 30 X 20). Number of days 
(ahargana) is multiplied by this 600 and divided 
by number of days in a yuga to give the no. of 
revolutions and fraction rasis etc. Its bhuja is 
multiplied by 3 and divided by 10, which will give 
ayana in amsa or ayanamsa. 

This gives an oscillatory motion of 27° east 
and west from equinox point. 



Corrections to Moon 345 

This appears a defective and interpolated 
passage because- 

(i) It occurs in Triprasnadhikara and out of 
context just after discussing directions and shadow 
lengths. 

(ii) No where else in this text krtya = 20 units 
has been used. 30 scores should have been written 
6 hundreds or each digit should have been 
indicated separately through words as per general 
practice. 

(iii) The verse indicates oscillation of naksatra 
cycle around equinox. If it starts with east ward 
motion; in 5097 years since kali, it should be 
towards west from equinox. But the 0° of ecliptic 
is towards east from equinox point, as it has been 
clearly mentioned in next verse also. Thus the text 
should have stated oscillations of equinox point 
around 0° of ecliptic. Due to 600 speed, round 
number (540) had been calculated at the beginning 
of Kaliyuga. 

(iv) Oscillations of equinox within 27° is not 
mentioned anywhere in ancient texts. They have 
mentioned the difference of upto 35° and values 
at different points of time indicates only backward 
motion. 

(v) Bhaskara II has quoted Surya siddhanta 
differently. According to him, surya siddhanta tells 
3 lakh backward rotations of Ayanamsa in a kalpa. 
This means 300 backward rotation in a yuga. This 
can mean 3 backward + 300 forward = 600 
oscillations in a yuga. But this interpratation has 
not been mentioned in own commentary or any 
other commentary. Thus he must have mentioned 



^ 46 Siddhanta Darpanak 

some version of surya siddhanta prevalent in his j 
time. This was lost due to the interpretation! 
presently found. J 

■ > 

(vi) Reasons of accepting this wrong version 
is that 0° position is same in both systems around 
285 AD. and both indicate backward motion till 
2298 AD. Due to approximately -equal angular speed 
in both system, we get the same position of 
Ayanamsa. So no body has thought it necessary 
to refute this theory. 

Reasons for oscillation theory and its value of 
constants- 

(i) Bhaskara and Varahamihira have com- 
mented that Ayanamsa was zero at the time of 
Aryabhata 3600 years exactly after Kaliyuga. Now, 
it has been assumed that all the planetary positions 
were zero at beginnig of Kaliyuga and they started 
moving east wards since then. The same assump- 
tion was made for krantipata which was found 
west from 0° at the time of Aryabhata. This means 
that, pata started moving east wards with uniform 
speed like all madhya grahas, at mid point till time 
of Aryabhata it started moving backward and 
reached zero position again. Thus half oscillation 
was completed within 3600 years. Remaining half 
oscillation will mean backward motion for 1800 
years from Aryabhata and again forward motion 
for 1800 years, so that it comes to zero position 
in east ward motion, as in Kali beginning. Thus 1 
cycle is 7200 years, giving 600 cycles in a yuga. At 
about 600 years after Aryabhata, if Ayanamsa was 
9° west, then maximum oscillation in 1800 years 
will be 27° on either side. Such measurement only 
can be basis of this limit. 



Corrections to Moon 347 

In comparison, Hipparchus (100 BC) had 
found precession but did not give the value. 
Ptolemy had estimated it to be 36" per year. 
Albatani of Arab in about 880 AD, found the speed 
as 55". 5 Then Nasiruddin of Iran calculated in 1250 
AD as 51" per year which was very accurate. 

Siddanta Darpana has assumed surya 
siddhanta theory of oscillation, but has slightly 
corrected the value to 6,40,170 oscillations in a 
kalpa instead of 6 lakhs for a kalpa according to 
Surya siddhanta. 

These corrections are based on the following- 

(i) Assumption of true 0° position which is 
with 1/2° error in eye estimates- This is according 
to position of identifying stars as given in Surya 
siddhanta. This will indicate, the current value Of 
ayananasa as to how much vernal equinox has 
shifted west from this 0°. 

(ii) Assumption about the time of 0° 
ayanamsa- it is clear that surya siddhanta value is 
based on ° ayanamsa at the time of Aryabhata in 
3600 kali in which half oscillation was complete. 
Figure of 6,40,170 oscillations in a year by siddhanta 
darpana indicates 0° ' ayanamsa in 284 AD. At 
present it is assumed to be on mesa sankranti of 
285 AD. So reasons of Candrasekhara must have 
been same as current reasons for accepting this 
figure. 

It may be noted that both theories give same 
figure at present because, their speeds are almost 
same. 27°X4 = 108° oscillation in 7200 years means 
1° in 66.6 years. Siddhanta Darpana gives 1° in 
61-4 years. Modern figure is 72.24 years per degree 
f or AD and 71.63 years at 1900 AD. Munjula 



348 ' «*■ ' Siddhanta Darpana 

figure also is 1° in 61 years. This was accepted by 
Bhaskara and this figure only has been accepted 
by Candrasekhar though under different theory. 

(3) Formulas explained : 

Revolutions of Ayana till desired day _ Ahargana 

Revolution in a kalpa " No. of days in a kalpa 

In a full revolution of 360°, quadrants are of 
90° each. In oscillatory motion the corresponding 
quadrants are 

8 - 90° 0° to + 27° 

90° --180° + 27° to 0° reverse motion 

180 ' - 270° 0° to-27° reverse motion 

270° to 360° -27° to 8 forward motion 

Thus 27° Ayanamsa = 90° revolution 

27 
or Revolution X — is Ayanamsa. 

27 3 60 

Hence revolution is multiplied by —= — or -r-r 

which has been mentioned here. 

Verses 92-99 - Calculation of Kranti 

Planetary orbits (ecliptic) and equator circle, 
both are in east west direction. Due to inclination, 
they cut each other which results in kranti (north 
south deviation). Thus, deviation of the planet, 
north or south from equator is measured along 
great circles passing through north pole and south 
pole (of earth projected in sky). This is also called 
'apama' or 'apakrama'. 

Note : Kranti (apama or apakrama) is north 
south deviation from equator ar seen from earth. 



'1 



■ J s 



Corrections to Moon ^49 

Sara or viksepa is north south deviation from 
ecliptic as seen from sun. 

Both are measured along great circle perpen- 
dicular to reference circle (equator or ecliptic). 

In celestial sphere, an imaginary circle of 
rotation of sun is called kranti vrtta or marga 
(ecliptic circle of path.) It is divided into 12 rasis. 
Ayana correction is done in 1st and 7th rasis (0° 
and 180° position). The corrected positions of these 
rasis give the positions of intersection of ecliptic 
with equator circle. These points are called pata. 
Since day and night are equal, they are called 
sampata. Thus there will be two sampata, vasanta 
and hemanta (vernal or autumnal equinox).. At 3 
rasis from sampata, kranti will be maximum (23° 30') 
in north or south directions. 

Jya of maximum kranti (23° 30') is (1370'). 
Graha position corrected by ayanamsa only is used 
for calculation of bhujaphala and jya. 

Ayanamsa is added to spasta graha, sum 
(sayana graha) is multiplied by Jya of parama kranti 
(1370) and divided by radius (3438). This is 
equivalent to multiplication by 100 and division by 
251. This will be kranti jya of the spasta graha. 
This value converted to arc will give kranti in kala. 
Square of kranti jya substracted from square 
of radius (1,18,19,844) and taking square root gives 
'dyujya' which is half diameter of ahoratra vrtta 
(diurnal circle) - explained in Triprasnadhikara. 

Kotijya of sphuta graha (corrected with 
ayanam§a) multiplied by 100 and divided by 251 
and multiplied by daily motion of graha gives daily 
motion of kranti. 



350 Siddhdnta Darpatui 

Ayana corrected graha moves northwards in 
1st and last quadrants and south wards in 2nd and 
3rd quadrants. 

Notes : (1) Kranti from sayana graha - 



'.:B 



A 0° P : 

O is the O" of ecliptic. By definition, kranti 
at point A of intersection of equator will be zero, 
because it is at equator also. A is towards west 
from due to backward motion. Planet P on ecliptic 
is counted in east direction from 0° of ecliptic. 
Thus kranti of planet P increases from A in the 
east direction, where it is zero. 

Thus sayana graha AP = OA (ayanarhsa) + 
OP (distance from 0° of ecliptic i.e. true graha). 

(2) As seen from equator, the pata, A where 
kanti vrtta appears moving north wards is the pata 
taken as 0°. 



Ectiptic 



■Wl 







'1 
■■■.1 



B 



■ h 



Equator 






J! 



Figure 5 | 

In figure 5, AE BA' is equator and AB' A' is j 

ecliptic which cut each other in line A O A'. OBJ 

and OB' are radius perpendicular to AA' at O 



-■VJf: 



Corrections to Moon 351 

which is point of observation at centre of celestial 
sphere. This equator and ecliptic are inclined at an 
angle B'OB which is about 23-1/2° 

Position of planet is at P on ecliptic whose 
distance from point A is the sayana graha = AP. 
PE is arc of great circle perpendicular to equator, 
hence passing through pole of earth or equator. 
Thus length PE is the kranti, which can be 
determined from relations of right angled triangle 
APE on the sphere. Hence sayana graha AP needs 
to be calculated to complete this triangle. 

According to Napier's law for right angled 
spehrical triangles, sine of middle part = product 
of cosines of opposite parts 

For middle part taken as PE, opposite parts 



are 



| - PA and | - ZPAE 

Hence sin (PE) = sin (PA) X Sin (PAE) 

~ v *. . - _ . _• RsinPA'x R sin PAE 
or Kranti lya = R sin PE = 

iJ R 

Here PA = sayana graha, Z.PAE = parama 
kranti 

Hence, krantijya 

= Jya of sayana graha x Jya of parama kranti 

Trijya 

Thus the position of highest kranti B' is at 
from A of 0° kranti. Another point of highest 
kanti is opposite to B' i.e. 90° from A'. 

s - Jya of parama kranti _ 1370 100 

Trijya = 3438 = 251 

a fixed quantity, hence alternate formula has 
°een given 



90 



352 



Siddhanta Darpan 



(3) Speed of kranti : 
Sin PE = Sin PA x 



100 
251 



. .'j* 



Differentiating both, cos PE. d (PE) = cos PA,] 

100 j 

d(PA).— 

For a single day, point A can be consider© 
fixed and d (PA) = d (AO + OP) = d (OP) = spee 
of nirayana graha 

as d (AO) = for small period 

PE is small and cos PE can be taken almos 
equal to 1. 

So speed of kranti is d(PE) 

100 

= — Xd (PA) cos PA 

= 152 X speed of eraha X cos of sayana graha 

251 ^ . • 

Verses 100-101 : 

According to Bhaskarcarya, ayana doesnl 
move in west direction, hence he has asked to ad^ 
ayanamsa to the graha always. Still according t 
Brahma and siirya siddhanta, I have assumed i\ 
motion in both directions. It will be clear by 
calculating ravi from chaya (shadow of gnomon). 

Verses 102-104 : 

Day night values at a place-Kranti jyi 
multiplied by palabha (shadow length of 12 len. 
stick on equinox day) and divided by 12 giv 
ksitijya. This, multiplied by trijya (3438) *i 
divided by dyujya gives carajya. Its arc will be 
prana. 

Caraprana added to \ of day night (15 danda| 
gives half day length when it is north kranti. Or 



an< 
car* 



Corrections to Moon 353 

abstraction from 15 dandas, half night length is 
obtained. When kranti is south, opposite procedure 
is followed - day half is obtained by 15 - caraprana 
and night half = 15 + caraprana. Multiplying them 
by 2 we get values of day and night (Quoted from 
surya siddhanta) 

For finding day and night periods of 
naksatras, moon and other planets, their sara is 
added to kranti, when they are in same direction, 
or difference is taken, when they are in opposite 
direction. From this spasta kranti, day or night 
time is found, (Day time is the period for which 
planet is above local horizon) 

Notes (1) These topics have been discussed 
in Triprasnadhikara, but to understand the mean- 
ing of these formulas, it is necessary to explain 
the technical terms. 

On equinox day, sun is perpendicular on 
equator, hence at local noon on an equator place 
it will be directly above, i.e. perpendicular to 
horizontal plane. Hence, a perpendicular to 
horizontal plane at other place with latitude 0, will 
be at an angle <p with sun's highest position at 
noon. Thus the length of a vertical pilllar's shadow 
at noon time on equinox day will give latitude of 
the place. 

N 

C A 



E' 




354 Siddhanta Darpan 



In Figure 6, S is perpendicular on equator; 
passing through E, S being direction of sun. Aif 
place P, latitude = 0. Hence direction of sun i 
CA direction makes ZCAP = <p with vertical 
direction of pole PA = 12 unit length. 




m 



Tan 6 = 



PC PC 
AC " 12 



...*J 



gives measure of latitude <p 




Figure 7 - Calculation of day time at a place 

Figure 7, is a diagram for place O where day 
length of a planet i.e. period for which it is above 
horizon is to be found. NOS is horizontal line in 
north south direction at that place and DOD' is 
the horizontal line for equator. D is celestial north 
pole (direction of earth's north pole in the sky) 
and D' is south pole of earth. N D V S is the 
north south circle and V is the vertically upward 
point at O. 

Due to daily rotation of earth, planets appear 
to move in circles parallel to equator. These circles 
are called ahoratra vrtta (diurnal circle). For 
different positions of a planet or naksatra, the 



Corrections to Moon 355 

circles projected on vertical plane are P1P1', P2P2 7 
and P3P3' all parallel to equator P2P2'. Sun on 
equinox day will appear moving on P2P2' - kranti 
for short time assumed constant. 

When north kranti of a planet is arc P2 Pi' 
then its diurnal circle is P1P1'. When south kranti 
is P2 P3, then the circle is P3 P3 7 . (diameter only 
is shown in projection). At equator, the horizontal 
line DOD' cuts all the diurnal diameters in two 
equal parts. As long as the planet is above horizon 
or on V side of DOD', it is seen or rising. Below 
it; it is set. Thus at equator, day and night are 
always equal. 

However, for place O, the horizontal line is 
SON. Day portion of the planet is PiH or P3H'. 
It is bigger than 12 hours for north kranti. 

OV is radius, PiK =Dyujya 

(diameter of diurnal circle) 

P1P2 arc or L PiO P 2 is kranti 

Hence, kranti jya = PiL = OK 

Versin of kranti = P2L (versin = 1- Cos 0) 

Dyujya = PiK = OL = OP 2 - P 2 L 

= Trijya - versine of kranti 

= Kranti koti Jya (1) 

Ksitijya = KH (extra motion on diurnal circle 
beyond half day). 

Latiude <p = Z.HOK or L VOP 2 

HK 

In AKOH, tan (j> = — 

OK 

_ Palabha 

But tan <p - —7Z — 
T 12 



356 Siddhanta Darpang^ 

„,. . . ,,„ Kranti jya x Palabha 
Hence Ksiti-jya KH ^^ - - (2) | 

But DKO and DHC both are perpendicular! 

on P1P1' and P2P2' (in the spherical triangle). I 

Pi K P 2 O 1 

Hence = 4 

KH OC 



nr nr- *** X F2 ° Ksi%a x Trijya ,§ 

or oc = ~ Sk = 3^T " " ■ (3 >l 

This is value of OC = carajya. Its angular! 
kala value is caraprana, because earth takes 1 prana | 
to rotate kala. 

From the equations (1), (2), (3), 

Carajya 

Kranti jya x Palabha Trijya 



12 Kranti Kotijya 

Krantiiya Palabha _ .. 

x „ x Tn jya 



Krantikotijya 12 
= Kranti sparsa jya X Aksansa sparsa X Trijya (4) 

In modern terms when Kranti is 6 

Sin (cara) = tan <p tan 6 (5) 

Complete day is rising from horizon H to top 
position and then coming back to M again, after 

1 

which it sets. Hence half day = — day night + 

carajya. 

Verse 105 : Correction due to sara in day time 

From surya siddhanta when kranti and sara 
are in one direction they are added to find spasta 
kranti of a planet (true declination from equator). 
When they are in opposite direction, their dif- 
ference is taken for spasta kranti. 

Notes : Kranti is inclination of planet from 
ecliptic. It is caused by two angles - Angle of 



Corrections to Moon 357 

ecliptic with equator whch is called kranti (mean 
value). However, a planet deviates from ecliptic, 
whose angle is known as sara. Hence total 
inclination with equator is sum of these angles. 
This inclination only, decides their day and nights. 

Verses 106-112 : Easy calculation of cara - 

Now a rough practical method is described to find 
out cara in pala. 

(i) Find out the cara kalas at the end of 1,2 
and 3 rasis (corresponding to their krantis) 

(2) 3rd cara khanda = 3rd rasi cara -cara of 
2nd rasi 

2nd cara khanda = cara of 2nd rasi - cara of 
1st rasi 

1st cara khanda = cara of 1st rasi itself 

■ • 

These are the cara of mesa, vrsa and mithuna 
rasis in reverse order. 

(3) Bhuja of sayana planets is taken, its rasi 
and degrees etc. are kept separately. If it is less 
than 1 rasi, then degree and minute (kala) are kept 
separately. 

(4) Degree and kala are multiplied separately 
by cara. Result at kala place is divided by 60, 
quotient in degree added to degree place, 
remainder to be kept as kala. Total degrees are 
divided by 30, remainder is kept there and quotient 
is added with rasi. 

If bhuja is more than 1 rasi; but less than 2 
rasi, position is multiplied by 1st cara for mesa 
rasi, kala and degrees are multiplied by cara khanda 
of 2nd rasi. As before, excess kala and degrees are 
added in higher places of degrees and rasi. 



358 Siddhdnta Darpanti 

If bhuja is more than 2 rasi (it will be always 
less than 3 rasi). then 1st and 2nd carakhandas are 
added at 1st place of rasi. Degrees and minutes 
are multiplied by 3rd cara khanda. These are 
converted to rasi, degree, kala as before. Alterna- 
tively, cara of each rasi of ravi is taken and 
accordingly, their fraction for each degree is 
calculated. 

Notes : Rationale of method is obvious. It is 
linear interpolation which assumes that variation 
rate of cara within a rasi (30° interval) is constant. 
This gives some error which can be ignored for 
practical purposes. 

Verses 113-117 : Udayantara pala from sayana 
surya - Now method to find udayantara sanskara 
is being explained. This is difference in pala 
between true sunrise time and madhyama sunrise 
time at Lanka. This is called time equation, arising 
out of inclination of ecliptic with equator. This rises 
steadily in first 3 half rasis (i.e. 3 X 1/2 X 30° 
= 45°) and decreases till next 3 half rasis. 

From the first sampata point, udayantara (in 
pala) rises by, 12, 9, 4 pala for first 3 half rasis, 
From 4th half rasi to end of quadrant it declines 
by same amounts 4, 9, 12. 

We find out the udayantara palas for com- 
pleted half rasis. Fractional portion of lapsed 
degrees is multiplied by pala of that 15° part and 
is added to the result for completed half rasis (if 
udayantara pala is rising). It is substracted if 
udayantara is declining. 



i 



: -a 



Corrections to Moon - 35 ^ 

When bhujansa of surya is 45° (3 X 15° or 
3rd half rasi), its udayantara pala is maximum 25 
palas (12 + 9 + 4 pala). After that it starts declining. 
On equinox day or at 4th rasis from that (0°,90°, 
180° or 270°) udayantara pala is zero. 

Udayantara pala is multiplied by daily motion 
of graha and divided by no. of pala in a day (3600). 
Result is added to madhyana graha, if bhuja of 
sun is in even quadrant, otherewise it is sub- 
stracted. Result will be the graha for sunrise time 
of Lanka. 

Notes : This is approximate udayantara palas 
at the end of each half rasis. Its complete 
explaination will be given in Triprasnadhikara. 
(p/447) 

Verses 118-120 : Rising time of planets - 

We add ayanamsa to graha, and from sayana 
graha its udaya time in asu (prana = 4 seconds) 
is found. Udaya asu is multiplied by daily motion 
of the planet and divided by no. of kalas in a rasi 
(1800) Result is added to kalas in a circle (21,600), 
if the graha is margi (moving forward). If vakri 
(retrograde), it is substracted from 21,600. The 
result will be day of the graha in asus i.e. after 1 
rise, it will rise after that time again. 

Savana dina for sun is roughly 60 danda. 59 
lipta less from that is a naksatra dina. Method for 
finding savana dina of a planet has been told. 

Notes : Udaya asu of a graha is its rising 
time, as its speed is seen from an inclined plane 
which will be less than its speed in the ecliptic. 
This will be less than its normal rising time. The 
corresponding apparant speed is found by dividing 




■■.a 
d 






Ml 






360 Siddhanta Darpa% 

the rising lime of that rasi by 1800 kala and 
multiplying it by gati of graha, this is movement 
in one day as seen from a latitude. If graha i$ 
moving ahead, this will be extra time taken 
earth to reach its next rising place. Hence this timelj 
is added to 21,600 asu. | 

Verses 121-126 : Rising time of rasis 

The rasi which rises (on eastern horizon) at 
a time is called lagna. At sunrise time, rasi, arhsa 
etc of surya itself is lagna. Savana day night 
(ahoratra) is found from daily motion of ravi as 
explained above. From true ravi at desired time, 
current rising rasi (lagna is found). 

In l/12th part of kranti vrtta (rasi), there are 
1800 kala. Near equator, their inclination to equator 
is more. At the end of ayana (south or north), i.e. 
90° east or west from equinox, kranti vrtta (ecliptic) 
is paralled to equator. Hence, in diurnal circle 
(parallel to equator), different parts of ecliptic rise 
in unequal times. 

To find out the rising times of rasis at equator, 
the jyas of 1,2 and 3 rasis and their kranti jyas 
are squared separately. For each rasi, kranti jya 
square is deducted from jya square, and square 
root of the difference is taken. These are multiplied 
by trijya and divided separately by jya of 1,2,3 
rasis. Arc of the three results is calculated. 

Rising time in asu for third rasi is found by 
substracting arc of 2nd from 3rd rasi. For rising of 
2nd rasi, arc of 1st is substracted from 2nd rasi. 
Arc of 1st rasi is its own rising time. 

Thus we get rising times in asu (udayasu) of 
(1) mesa (2) vrsa and (3) mithuna rasis. Udayasu 



Corrections to Moon 361 

of next 3 rasis are in reverse order i.e. rising time 
of (4) karka is same as of 3rd rasi, of 5th sirhha 
it is same as of 2nd rasi and of 6th kanya and 1st 
is same. 

Rising times of tula to mina is in reverse order 
of the times for 1st to 6th rasis. 

Comments (1) Steps in calculation 

Jyas of the rasis (1,2,3) or 30°, 60°, 90° are 
1719, 2978 and 3438 

Their squares are (29,54,961), (88,68,484) and 
(118, 19,844) 

Kranti jya of 3 rasis are (685), (1186), (1370) 

Their squares are (4, 69, 225), (14, 06, 596) 
and (18, 76, 900) 

Substracting kranti jya squares from jya 
squares, we get (24, 85, 736), (74, 61, 888) and (99, 
42, 944). 

Square roots of these results are 

(1576/37), (2731/39) and (3153/15) 

They are multiplied by trijya (3438). Products 
are (54,20,408), (93, 91, 413), (108, 40, 873) 

They are divided by respective dyujyas 

(3369), (3227), (3153) 

Results are (1609), (2190) and (3438) 

Their arcs are (1675), (3471), (5400) 

Rising time of mesa = 1675 

Rising time of vrsa = 3471-1675 = 1796 

Rising time of mithuna = 5400-3471 = 1929 

These are better approximations for modern 
values for 23° 27' inclination of equator. These are 



362 Siddhanta Darpana 

based on 23 ° 30' declination and old siddhantas 
assumed 24°. Comparison is given below 

Modern 
values 



Sayana 
rasi 



Rising time 
in asu in 
old siddhanta 



Rising time 
in Siddhanta 
Darpana 



Mesa 
Vrsa 

* • 

Mithuna 



1670 
1795 
1935 



1675 
1796 
1929 



Asus Minutes 

1675 111.7 

1794 119.6 

1931 128.7 



Lanka rising time for all rasis (Siddhanta 
darpana) 



Value 
1675 
1796 
1929 



rasis 



rasis 



rasis 



rasis 



(1) Mesa (6) Kanya (7) Tula (12) MIna 

(2) Vrsa (5) Simha (8) Vrstika (11) Kumbha 

(3) Mithuna (4) Karka (9) Dhanu (10) Makara 

(2) Derivation of rising time formula for 3 rasis 




Figure 8 - Rising times of rasis at equator 

Figure 8 is horizon circle of equator in which 
E, N, W and S are the points in east, north, west 
and south. 

WOE is equator circle 
K'OK is ecliptic projection 
. O = Vasanta sampata (or vernal equinox) 



Corrections to Moon 363 

N is also direction of north pole of earth. 
Daily rotation of earth is along circle WOE, the 
time in which OE part of equator rises, is the time 
of rise of OK part of ecliptic also. But rising time 
of the whole equator circle 360° is 1 naksatra dina 
(sidereal day) which is equal to 21,600 asus by 
definition. Hence rise of 1 kala on equator will 
take 1 asu. Hence length of OE in kala will give 
the rising time in asu which is rising time of OK 
part of ecliptic also. 

OEK is a spherical triangle in which ZOEK 
is right angle, ZEOK is angle between equator and 
ecliptic which is maximum value of sun's kranti. 
EK is kranti of point K, arc OK is sayana rasi of 
point K measured from equinox point O. OE is its 
length measured on equator (visuvansa). 

Hence as per Napier's rule - 
Cos KOE = tan OE x cot OK 

cos(parama kranti) 

or tan OE = ——. -,. N - - \\) 

cot (sayana rasi) 

For finding values in R sines (jyas), relations 
in spherical triangle NOK, 

SinNK _ SinOK 

sin NOK " sinONK 

But L ONK = arc OE 

SinNK sin OK 



Hence 



sin NOK SinOE 

sin OK X sin NOK 



or Sin OE = . XT1 , 

sin NK 

Here OK = sayana value of K 



364 Siddhdnta Darpana 

L NOK = ZNOE - L KOE = 90°- Parama 
kranti of sun 

Hence Sin NOK = cos (parama kranti) = 
Dyujya of 3 rasis 

(because cos (kranti) = Dyujya) 

Sin NK = Sin (NE - KE) = Sin (90 ' - kranti 
of K) 

= cos (kranti of K) = Dyujya of K 
Thus Sin OE 

Sin (sayanaK) x Cos (Parama kranti) 

cos (kranti of K) * ' 

Alternatively it is, Sin OE 

Sin (sayanaK) x Dyujya of Parama kranti 

Dyujya of K (3) 

Formula (3) has been given in the next verse. 
In . spherical triangle KOE 

sin KE sin OK 

~ — ix^TT = ~ — ^T^7 = sin OK ( as sin OEK = 
sin KOE sin OEK v 

sin 90° =1) 

Thus in formula (2) 
Sin OE 

Vl - Sin z KOE 



= Sin OK x 



D yu jya of K 



sin OK , _ 



v 



2 



1 sin Ke x 
sin OK 
K Dyujya 

Vsin 2 OK - sin 2 KE 
K dyujya 

or R sin OE= V (R sin Ok)' - (R sin KE)* 

Dyujya of K y) 



Corrections to Moon 365 

This is the formula described in this verse. 

(3) To prove that rising times of 4th to 6th 

rasis are equal to those of 3rd to 1st rasis in reverse 
order - 

Equation (3) above tells 

sin OK x cos (Parama kranti) 

Sin OE = ^=t 

cos (KE) 

OE = rising time or length on equator in kala. 

OK = sayana rasi of K, KE = Kranti of K. 

Sin = Sin (180°- 0) 

Hence Sin (180° - OE) 

Sin (180° - OK) x Cos (parama kranti) 

cos KE 

Rising times of 90° at equator or ecliptic are 
same i.e. when OK = 90% OE == 90°. 

For rising time of mithuna (60 o -90°), we 
substract the rising time of 60° from 90° time (6 
hours = 15 danda = 5400 asu). 

Rising time of 180° also is equal on both circles 
as it is equal for every 90 ° . Hence, rising time for 
karka (90° to 120°) is found by substracting the 
time of rising time of 3 rasis from 120° time. 

Now, when OK = 60°, OE is rising time 
(slightly less than OE) 

When OK = 120° = 180° -60°, its rising time 
= 180° - OE 

Hence rising time *>f Karka = (180°-OE) - 90° 
= 90' - OE = rising time of mithuna. 

Similarly we can prove that rising times of 
simha, vrsa and kanya, mesa are equal. 



366 Siddhdnta Darpana 

The rising times of rasis from mesa to kanya 
are equal to tula to mlna in reverse order for all 
palces, not only on equator. So this result will be 
proved when rising time at other places is 
calculated. This is evident because both the ecliptic 
and equator circles bisect each other, hence other 
half 180° to 360° is similar to 180° to 0°. 

Verses 127-128 : Alternative method for rising 
times at equator 

Dyujya of 3 rasis (3153) is multiplied separate- 
ly by jya of 3,2, 1 rasis (3438/2978/1719). Results 
(10,840,014), (93,89,634), (54,20,007) are divided by 
dyujya of the rasis (3153, 3227, 3369). Arc of the 
resulting ratios treated as jya is found (5400, 3471 
and 1675), which are rising times of 3, 2 and 1 



rasis. 



Rising time of 2 rasi is substracted from 3 to 
give time of 3rd rasi. Time of 2nd rasi is time of 
1st rasi deducted from rising of 2 rasis. Rising time 
of 1st rasis is already known. 

Notes : This method has already been proved 
in previous verse. 

Verses 129-130 : Rising times at different parts 
of sky. 

Rising times of six rasis in asu or prana have 
already been stated as (1) 1675 (2) 1796 (3) 1929 
(4) 1929 (5) 1796 and (6) 1675. (These have been 
calculated for rising on east horizon on equator). 
The rising time of rasis for other points (on the 
east west vertical circle) are also the same. These 
points are udaya (east horizon), Asta (seetting point 
in west horizon), Dasama (Tenth house or vertically 



Corrections to Moon 367 

upward point), Caturtha (fourth or vertically down 
wards). 

Note : This is because all quadrants are same 
on both circles. 

Veerses 131-142 : Lagna at any place - 

To find .out rising times of rasis at other 
places, we find out the cara khanda of first three 
rasis as per formula described for that place. These 
cara khandas are deducted from first and last 3 
rasis in that order and are added to the three rasi 
from karka and in reverse order to three rasis from 
tula. Addition and deductions of carakhandas is to 
the rising times of rasis at Lanka (both in asu or 
prana). These give the rising time at other place, 
for which carkhanda had been calculated. 

• ■ 

According to rough calculation, whatever rasi 
is rising in east horizon, its seventh rasi (180° 
away) is setting in the west. 

Rising times for each hora (1/2 rasi = 15°), or 
dreskana (1/3 rasi = 10°) can also be calculated in 
same manner. For that kranti and dyujya is 
calculated for each half or l/3rd rasi, hence it will 
be more accurate, than rising time for rasis. 

Ayanamsa is added to sun at sunrise time 
position. Lapsed and remaining parts in the 
incomplete rasi of sayana sun is calculated. Remain- 
ing degrees of the rasi are multiplied by rising 
time of full rasi and divided by 30. This gives 
rising time of remaining part of that incomplete 
rasi. This is substracted from desired time interval 
after sunrise (called ista kala). From the remainder, 
rising times of next rasis in successive order are 
deducted. Last remainder from which rising time 



''■'■■$ 



Siddhdnta DarpcajM 



368 

of next rasi cannot be deducted - is multiplied by 
30 and divided by rising time of the next rasi. This ? 
result in degrees etc. is added to the completed > 
rasi which has risen. This gives sayana lagna. 
Ayanamsa is deducted from this to give the lagna 
for required time at desired place. 

When fractional rising time of remaining rasi 
of sayana sun is more than ista kala, the same 
sayana lagna will continue to rise at ista kala. This 
remaining rising time is multiplied by 30 and 
divided by rising time of sayana sphuta ravi (or 
roughly by rising time of that rasi). Result in 
degrees etc is added to sayana sphuta sun and 
ayanamsa is deducted to find sphuta lagna. 

To find the moment when a particular lagna 
will rise, ayanamsa is added to it. Its lapsed part 
in incomplete rasi is multiplied by rising time of 
that rasi and divided by 30. This is lapsed rising 
time of the fractional rasi. To this, we add the 
rising time of remaining fraction rasi of sayana sun 
at sun rise time, and the rising times of next 
completed rasis upto the completed rasi of sayana 
lagna. The grand total will be ista kala after sunrise, 
when the desired lagna will rise. 




Wk £ 



Figure 9 - Rising times of risis at places other than equator 



Corrections to Moon 3&g 

Note : (1) Rising times for a place of latitude 
<p NESW is the horizontal circle at desired place 
of latitude q north, (fig. 9) 

P is the north pole in sky : 

O is vernal equinox point. WOE is equator 
circle, KOK' is ecliptic circle. EC = Cara of K which 
is below horizon. 

When point O is rising on east horizon, 
sayana 0° of both ecliptic and equator are rising. 
When K point of ecliptic rises on horizon, E point 
on equator also rises. 

Hence, rising time of OK in asu is same as 
that of OE. The polar circle PK passing through 
K meets OE extended at C which is below horizon. 
Thus OEC arc is the rising time at equator for 
point K. Hence rising time at 0° North is found 
by deducting EC from rising at equator. EC is the 
cara-kala for point K. 

Thus rising time = Equator rising time - 
carakala. 

Cara jya = R tan <p x tan 

where is kranti of K. It has already been 
proved after verse 103 

For mesa rasi, OK = 30° 

Kranti of K is KC 

Carajya = R tan KC X tan <p = EC 

Rising time OE = OC - EC 

This holds good for mesa to mithuna i.e. 0° 
to 90°. For karka rasi, OK = *120\ Then kranti of 
K is same as of vrsa rasi i.e. KC is same. Hence 
EC is also same as for 60° (vrsa). 

Hence rising time of karka = OE = OC - CE 






'"■ * k 



370 Siddhdnta Darpa$§ 

=(Rising time, of 3 ra&s + karka) - cara otj 

Vrsa 

= (Rising of 3 rasis-cara of 3 rasi) + karka + 

(cara of 3 rasi - cara of vrsa) 

= (rising time of 3 rasi at lat) + karka + j 
cara of mithuna 

Hence extra rising time for karka = karfc*| 
rising at equator + mithuna cara. | 

Similarly cara of vrsa is added to sirhha, ami | 
mesa cara is added to kanya rising time at equator.^ 

E 1 




i 






1 



A 



rH 



Figure 10 - Rising limes for tali to mira 

(2) Rising times for rasis tula to mina - Tg 
fieure 10 for 2nd half of ecliptic is same but the 
difference is that the two circles after crossing ead* 
other at autumn eqiunox O, have .« versed a *^ 
positions. K'O part of ecliptic which was abovj 
equator till 180' at O sayana, goes below equate* 
after O at OK i.e. after sayana tula. Hence, tui* 
to 3rd rail from it, cara portion CE is to be a 
to the rising times at Lanka. Thus ^."""SJT 
= tula time at equator + cara of 1st 30 (mesa}. 



i 




1 



Corrections to Moon 371 

This is same as rising time of kanya as proved 
in previous section. 

Similarly times of vrscika and simha are same 
and so on in that order. 

(3) Calculation of lagna - OE is east horizon 
at sunrise time and OE' is its position at ista kala 

after sunrise. Ai, A2, A3 A7 are successive 

positions of starting of rasis. (fig. 11) 

At sun rise time, point E is on east horizon 
and lagna, and sun also is rising at E. Hence, at 
sunrise, rasi of sun and lagna is same. The rising 
time of E' is the sum of rising times of EA2 
(remaining part of fractional rasi A1A2) then rising 
of complete rasis A2 A3, A3 A4, A5 A6 and then 
lapsed part of fraction rasi Ae E'. 

A . 1 E A2 A3 




Figure 11 Calculation of Lagna 

Within a rasi the rising times can be con- 
sidered as proportional to the parts, hence 

Rising time of E A2 _ degrees of EA 2 

Rising time of Ai A2 30 ° of Ai A2 

Similarly rising time of completed part A6 
E'can be calculated as fraction of rising time of rasi 
AeA 7 . 



372 Siddhdnta Darpm 

Since the rising times are not proportional t 
rasi length (mesa rising is much faster than vrs 
for example) this calculation will be more accurate, 
if rising times of smaller parts like hora = 1/2 
or dreskana = 1/3 rasi are calculated. 

• ■ 

Verses 143-151 : Rule for finding das 
lagna : Madhya lagna or tenth lagna (vertical to 
position) is found by rising time of rasis at equat 
only for all places. (Because south north line bisec 
the diurnal circles at all places and corresponding 
times are same at any place and equator). 

Before mid-day, the period for which sun 
remain in east is called nata kala which is desire 
time before mid day. 

Nata kala in east direction is expressed in asui 
From this, we deduct the equator rising time o J 
completed part of sayana sun rasi. From remainder! 
the equator rising time of previous rasis i' :J 
substracted successively. Last remainder is divide 
by equator rising itme of next rasi (whch cann 
be deducted from remainder) and multiplied b; 
30. Result in degrees etc is substracted from 30" 
This is added to the previous rasi which become 
sayana madhya lagna. Spasta madhya lagna 
found by deducting ayanamsa. 

When nata kala is west, the time passed aft 
mid day is nata kala. From this we deduct the 
fractional rising time of sayana sun at equator. 
Then rising times of next rasis are deducted. Last 
remainder is divided by rising time of incomplete 
rasi and multiplied by 30. Result in degree etc. is 
added to completed rasis to give sayana madhya 
lagna. From this, ayanamsa is to be deducted. 




Corrections to Moon 373 

When in purva or pascima nata, nata kala is 
less than the rising time of fractional rasi (lapsed 
or remaining, then nata kala is divided by rising 
time of the rasi and multiplied by 30. Result in 
degrees etc is substracted from sayana ravi for 
purva nata and added to it for pascima nata. 
Sphuta sayana surya at midday is the madhya 
lagna at that time. 



pi 



w 





P3 




L^P4 




P2 






V N 


X P5 


B r 






■ 


'Aa 













B' 








*A" 



P6 



Figure 12 

Notes : (1) Figure 12 is the vertical circle of 
any place O. E and W are east and west points. 
T is the top most or vertically upwards point. D 
is opposite to T and down ward point. Earth is 
rotating in clockwise direction, hence ecliptic 
appears moving in anti clockwise direction - shown 
by arrow. 

Movement of ecliptic is not in this plane and 
°nly its projection is considered. At sunrise time, 
its projection will be at E which is not at 90° from 
D, it is upwards for north latitudes when kranti 
°f sun is south. ETW is the day position and WDE 
is night portion of sun. DET is position of purva 
n ata (mid night to mid day) and TWD is pascima 
nats. Vertical point T is same for all places, because 



374 Siddhanta Darpan 

apparent rotation wil be paralel to equator. Heri 
rising times at equator are taken. 

For a position of sun at A in purva nata, thj 
tenth lagna is lagna at point T. Pi,P2 - - - P6 ajp 
the start of successive rasis which will rise on 
after other in clockwise direction. Thus the rasi 
sun at A will reach T after travelling AT portioj| 
Current rasi at T is less than A. For point B or WM 
at pascima nata, rasi at B has aleady risen at T % 
hence current rasi is more than sun's rasi at ; B. | 
Position of sun at A or B indicate the time. 

Thus, for purva nata of sun at A, tenth lagna 
at T is = A - AP 5 - P5P4 - P4T 

For pascima nata, Sun is at B, tenth lagna at ; 
T is = BP 2 + P2P3 + P 3 T. 

For calculation of rising time of part rasis, the 
rising time is considered proportional to degrees 
within the rasi which is roughly correct. 

Verses 152-153 - Rising time of Nirayana rasis 

- when ayanamsa is moving eastward, we take the 
difference of rising times of desired rasi (sayana 
value) and next rasi, it is multiplied by ayanamsa 
and divided by 30. If rising time of next rasi is 
more, then the result is added to rising time of 
sayana rasi to get the rising time of nirayana rasi. 
If next rising time is smaller, it is subs tr acted. 

When ayanamsa is moving west wards (which 
is not the current position), we take the difference 
of rising times of the desired rasi (sayana value) 
and the rising time of previous rasi, it is multiplied 
by ayanamsa and divided by 30. If the rising time 
of previous rasi is more, result is added to sayana 



Corrections to Moon 375 

rising time to get the rising time of nirayana rasi. 
If previous rasi time is more, it is added. 

Notes : Method is obvious. Since ayana is 
moving east wards, sayana rasi is more. Hence, 
rising time of nirayana rasis will be found by 
comparison with the next rasi. Assuming ayanamsa 
of 23°, nirayana mesa 0° = sayana mesa 23°, 
nirayana mesa 30° = sayana mesa 53°. Hencewe 
have to find the rising times at 23° and 53° at 
sayana value, and their difference is rising time 
for mesa. 

Verse 153 - When ayana (ecliptic) is moving 
west wards (from point of equinox) then rising 
times of cara rasis (1, 4, 7, 10th rasis) is same for 
sayana and nirayana values at equator. At other 
places also rising times of mesa and tula will be 
same for sayana or nirayana values (Their previous 
rasis have same values). 

When ayana (ecliptic) is moving east from 
equinox point, then dvisvabhava rasis (3, 6, 9, 12th) 
rasis have same sayana and nirayana rising times 
at equator. At other places, only the 6th and 12th 
rasis have same rising times for sayana and 
nirayana (as the rising times for next rasis are 
same). 

Verses 154-156 : Rising times for Orissa (22 °N) 
and equator 

Rising times for sayana rasi at middle of 
Utkala (22 °N) in danda pala etc are as follows - 

mesa (3/56), vrsa (4/24), mithuna (5/7), karka 
(5/37), simha (5/34), kanya (5/22). For second half 
circle starting from tula, values are in reverse order. 



w 



~ • • 



«376 Siddhanta Darpatut 

Udaya times for 22° Ayanamsa - for nirayana 
rasis are 

mesa (4/17), vrsa (4/56), mithuna (5/29), karka 
(5/35), sirhha (5/25) kanya (5/22), tula (5/31), vrscika 
(5/36), dhanu (5/15), makara (4/35), kumbha (4/3), 
mina (3/56). 

Nirayana rising times at equator are mesa 
(4/54), vrsa (5/16), mithuna (5/22) karka (5/35), simha 
(4/44), kanya 4/39), 

Values from tula etc will be in reverse order. 
There values will change with change in ayanamsa. 

Veerses 157-160 - Values and Charts 

Values of 28 naksatras have been stated here 
according to sages like Garga and Vasistha. Extent 
of rasis (30°) is clear. Dhruva (constants) have been 
stated in lipta approximately. Motion of pata of 
equator and ecliptic also have been given in 
appendix for 73 values of days. From them, 
ayanamsa for any day can be calcualted. 

Motion of pata of equator and ecliptic in a 
day is liptas etc 0/0/31/32/51/35/6/53/28/23. 

On first day of kali, visuva sampata was 702 
liptas from fixed mesa 0° (towards east). 

In Karanabda beginning, kranti pata in rasi 
etc. (for 1869 AD) was 3/16/33/47/27/40. 

To find out the ayanamsa since kali beginning 
easily, we find the years since kali beginning 
according to madhyama siirya. (219/19) is deducted 
from it. Result is multiplied by 100 less (i.e. 2416) 
and divided by 2516. We get result in kalas. By 



'4 



Corrections to Moon 377 

dividing with 60 we get degrees. Hara of 54° is 
substracteed from it. 

Notes : (1) Kranti pata and ayanamsa are 
considered same thing. But in terms used in the 
book, kranti pata always moves in reverse direction, 
making complete circles of 360°. But ayanamsa is 
3/10 of its bhuja calculated according to quadrant 
of the kranti. 

Kranti at kali beginning can be calculated by 
multiplying kalpa bhaganas by 1811 and dividing 
by 4000. (chapter 3, verse 51). Thus bhagana at 
kali beginning 

1811 
= 640170 x -— — = 2,898,36. 9675 revolutions 

4000 

This is 0.0325 revolutions less than complete 
revolutions. 

Since pata in moving in backwards direction, 
it is 0.0325 east of 0°. Thus Kali position is 0.0325 
revolutions = 0.0325 X 360° X 60 lipta = 702 liptas. 

One revolution is in 432 crore years of kalpa 
divided by 640170 revolutions i.e. 6748.207507 
years. 

Hence pata will come to 0° in reverse motion 



in 



6748,207507 m 
360 x 60 

19 ., 
= 219 — Years approx. 

60 



Hence revolutions at kali beginning are 
counted from 219/19 yeears after kali (position of 
0°). In Karanabda 4971 years had been completed 
since Kali. In about 3374 years half revolution will 
be complete in 3374+219 = 3593 kali year. Remain- 



378 Siddhdnta Darpana 

ing years are 1378 years in which less than 1 
quadrant will be covered. Thus with reverse motion 
pata crossed 4th and 3rd quadrants in half cycle 
and is now in 2nd quadrant at the end of which 
(i.e. at end of 1st quadrant in forward motion), 
kranti will be 90° correspondign to - 90° pata or 
+ 27° ayanamsa. 

By this method the Karanabda pata is about 
3 Y 16° 34' approx. Ayanamsa is less than 3rd rasi 
position of 27° by 16° 34' x 3/10 i.e. 27 '-4° 58.2' = 
22 °1' approx, which is given at the end of verse 

83. 

(2) Ayanamsa = (y-210/19) X 2416/2516 Kala X 
1/60 degrees. 

219/19 years are deducted because, in that kali 
year kranti pata and ayanamsa were zero. As the 
ayanamsa had become zero after 1/2 revolution of 
kranti 27°X2 = 54° movement of ayanamsa in 2 
quadrants, this amount is substracted, called hara. 

. . ,. 108 x 60 

Movement per year in kala = ,„ Mn „ MM „ 

^ J 6748.207507 

= 0.960255 Kala 

0.960255 X 2500 = 2416.0016 

2416 
Hence annual movement of ayanamsa = Kalas 

Thus we get the formula. 

Verse 160 - Charts have been given in 
appendix for Jya (R sine) for 24 khandas of 3 rasis 
beginning from mesa, kranti in kala, semi diameter 
of diurnal circle, carkhandas in asu for Purusottama 
Ksetra (Puri), rising times of rasi at equator in asu 
and udayantara phala in asu for convenience of 
students. Intermediate values can be known by 
proportional increase. 



Corrections to Moon 37 ^ 

Verses 161-162 - Prayer and end - 

Supreme lord had directed Brahma to create 
grahas for knowing the earned karma of previous 
births and fate in the present birth. Brahma, in 
turn, regulates motion of planets through sighra, 
manda, pravaha, pata etc. The same supreme Lord 
Jagannatha has created Purusottama Ksetra for 
emancipation of beings. I pray that Lord Jagannatha 
living at NUacala. 

Thus ends sixth chapter describing kranti, 
accurate sun and moon etc in Siddhanta Darpana 
written by Sri Candrasekhara born in a bright royal 
family of Orissa, for purpose of educating students 
and for tally in calculation and observation. 



Chapter - 7 

THREE PROBLEMS OF DAILY MOTION 

(Triprasnadhikara) 

1. Scope - There are three problems regarding 
daily motion of earth, or rather it is used to find 
their answers - 

(1) Place - Longitude or Latitude can be 
determined from daily motion. Both are needed to 
find the location of a place, specially in sea journey, 
when there is no other land mark for identification. 

(2) Direction - North south direction can be 
measured roughly by a magnetic compass also, 
which gives other directions also. But this causes 
a lot of errors, because magnetic north pole is 
different from geographical north pole, which is 
on the axis of earth's rotation. In addition, there 
are local and general magnetic disturbances. Ac- 
curate method of finding the directions is only by 
astronomy, whether on land or on sea. 

(3) Time - Measurment of time intervals are 
most accurate now with quartz watches for common 
use and most accurate laser and atomic watches 
for scientific use. However, that gives average 
standard time. True or apparent time can be found 
only by inclination of sun from vertical position. 
This is related to measurement of longitude also, 
as simultaneous measuring of time through sun at 
two places will be different, the difference depend- 
ing on longitude. Thus time difference or Ion- 



Three Problems of Daily Motion 351 

gitudinal difference can be calculated from each 
other. 

Siddhanta Darpana has treated this chapter 
in briefest manner and one of the vital use i.e. 
measurement of longitude has been left out. It has 
been explained roughly for purpose of making 
desantara correction in madhya graha in chapter 
4. One reason of such neglect is that use of 
astronomy for navigation had ceased for Indians, 
who had lost the traditional excellence. This doesn't 
mean that astronomy is not needed for this purpose 
now. Even in modern astronomy, exactly the same 
methods are used for finding directions, place and 
time. With use of telescopes, their accuracy has 
increased, but formula is same. 

Another reason for leaving some topics has 
been stated by the author that many more methods 
have been explained in detail by Bhaskaracarya, 
whose book is most popular. Hence they need not 
be repeated. Before explaining individual methods, 
it will be useful to give a general idea of various 
right angled triangles used for calculations. 

(2) Latitude triangles 

For calculation of 3 problems, some convenient 
right angled triangles are formed, whose one of 
the angles is latitude or aksamsa . Hence they are 
all called latitude triangles or 'aksaksetra' in Indian 
astronomy. The other angles of such triangle are 
obviously 90° - <p, and 90° as it is right angled 
triangle. 90° - <p is called colatitude or lambamsa. 
The side facing angle <p is called base (bhuja or 
bahu), side. Facing 90° - (p is upright (koti) and 
the side facing right angle is hypotenus (karna). 



382 



Siddhdnta 










The radius R of the celestial sphere is assumed tqfc 
be 3438 or, more correctly 3437'44" (which is value 
of one radian). 

(1) Let S be the sun (or any other heavenly 
body) on the celestial sphere at any given time, 
SA be the perpendicular dropped from S on the 
plane of the celestial horizon, SB the perpendicular 
dropped from S on its rising setting line and AB 
the perpendicular from A on same line RT. (R is 
rising point on horizon and T is setting point). 

OS = R, altitude of 
S is L SO A = a. Hence 
height of S = SA 
= R sin a is the sanku. 
SB (hypotenus) is called 
'Istahrti'. It has beenw 
called' 'dhrti', 'svadhrti' T 
Istadhrti', 'nijadhrti' etc. 

AB is called 
'sankuntala' or 

'sankvagra' 

Figure 1 

L AS B = <p, hence AASB is a latitude triangle. 
In this 

Base upright Hypotenus 

Sankutala Sanku or R sin a Svadhrti or Istadhrti (1) 

(2) When S is on prime vertical, SA is called 
sama - sanku' AB 'agra' and SB 'samadhrti' or 
tad-dhrti'. 

Base Upright Hypotenus 
Agra Samasanku taddhrti (2) 

(3) When S is on prime vertical, then if a 
perpendicular AC is dropped from A on taddhrti 




Three Problems of Daily Motion 3S3 

gB two more latitude triangles ACB and ACS are 
formed, AC = R sin d where <3 is declination. CB 
is called earth sine (ksitijya), kujya, Bhujya or 
mahajlva etc.) SC = taddhrti - kujya 

Base Upright Hypotenus 

Earth sine R sin d Agra (3) 

R S in d taddhrti Samasaniku - - - (4) 

(4) When Sun is on the equator and S its 
position on the celestial sphere at midday, SA is 
perpendicular on the plane of celestial horizon and 
O is centre of the celestial sphere, then SAO is 
again a latitude triangle. Then L OSA = <p 

Base Upri^t Hypotenus 
Rsin0 Rcos0 R (5) 

(5) When Sun is on the equator, then at 
midday, the gnomon, (a vertical pillar of 12 unit 
length called sanku), its shadow (equinoctical 
midday shadow - palabha, aksabha, palacchaya 
visuva chaya etc) and hypotenus of the mid day 
shadow(called palakarna, palasravana, akskarna, 
aksasruti etc.) also form a latitude triangle. This is 
called fundamental triangle and has been explained 
in previous chapter for calculation of lagna, day 
time etc. 

Base Upright Hypotenus 

Palabha gnomon or 12 palakarna - - - (6) 

Then there are two altitude triangles for sun 
Base Upright Hypotenus 

(1) Sanku drgjya or 

(R sin a) natajya R v) 

(R sin Z) 

(2) Gnomon or 12 shadow Hypotenus of (8) 

shadow 



'.. -is 



384 Siddhanta Darpana 

(3) When the sun is on the meridian, sanku 
is called 'madhyasanku' or madhyahna sanku'. 
Shadow is called 'madhyahna chaya karna'. 

(4) When the Sun is on the prime meridian 
(Samamandala), sanku is called sama sanku, 
shadow is samacchaya and hypotenus of shadow 
is samacchaya karna. 

Translation of the text 

Verse 1 - Scope - For happiness and benefit 
of the people, I begin this chapter named 'triprasna' 
which will give knowledge of dig (direction), desa 
(location) and kala (time) in simple language. 

Verses 2-5 : Finding the cardinal direction 

To determine the directions, a place is made 
plain like a surface of water. It is cleaned and a 
circle of semidiameter 24 angulas is drawn. At 
centre a Sanku of 12 angula height is kept. 

Shadow of sanku will touch the circle twice 
(when its length is 24 angulas). Both points on 
circumference are joined by a line and with each 
point as centre, circles of 25 angula radius is drawn. 

Both the circles will intersect at two points 
and common parts of circles between them will 
form a fish shape. The points are like mouth and 
tail of the fish. The line joining them will be north 
south line which will be perpendicular on the line 
joining shadow position between first and second 
chaya. Kranti movement is negligible and is 
ignored. 

North south line will cut the circumference 
on two points called north and south points. A 
perpenducular on that line at the centre will cut 
the circle in east and west points. For finding 



■;-i 



Three Problems of Daily Motion 3S5 

angular directions, arcs between east, north, west 
and south are bisected. 

Notes (1) Types of sanku - Bhaskara I in his 
commentary on Aryabhatiya has described the 
following views - 

Some astronomers prescribed a gnomon 
(sanku), whose one third in bottom is in shape of 
a prism on square base (caturasra), one third in 
middle in shape of cow's tail and one third in the 
top in shape of spear head. 

Some other have prescribed a square pris- 
moidal gnomon. 

The followers of Aryabhata I, used a broad 
(prthu), massive (guru) and large (dirgha) cylinderi- 
cal gnomon, made of excellent timber and free from 
any hole, scar or knot in the body. 

For getting the shadow ends easily and 
correctly, the cylinderical gnomon was surmounted 
by a fine cylinderical iron or r wooden nail fixed 
vertically at the centre of the .upper end. The nail 
was taken to be longer thanivthe radius, jof the 
gnomon, so that its shadow was always seen on 
the ground. 

(a) Height of gnomon - 

Gnomon could be of any length, but its height 
was divided into 12 units by convention. Smallest 
was gnomon of 12 angula length, because it was 
portable and easy to handle, (about 9.8" = 24 cm). 
Whatever may be length, it was called 12 angula 
marked by 12 equal division which will be cieariy 
seen in the shadow. Angula also was divided into 
60 pratyahgula for accurate measurement. It may 
be mentioned that accurate measurements were 

*7- 



* <?*■■ -■■ <" 



386 Siddhanta Darpi 

based on very long gnomons. Visnudhvaja 6- 
Kutubminar at Delhi was one such pillar. Sinci 
this indicated or marked (like a flag or dhvaja) th 
position of sun (Visnu) it was called visnu-dhvaja 
Its arabic translation means the same thing, Kutu 
means north south direction (Kutub-numa=com 
pass) manar or minar is measurement or tower fa 
that purpose. 

(b) Testing the level of ground - 

Test prescribed by Bhaskara I, Govinda Sva 
and Nflakantha is - 

• • 

When there is no wind, place a jar of wat 
on a tripod on the ground which has been mad 
plane by means of eye or thread, and bore a (fine 
hole at the bottom of jar, so that water may hav 
a continuos flow. Where the water falling on th 
ground spreads in a circle, there the ground is 
perfect level. Where water accumulates, it is low 
It doesn't reach at high level. 

The same principle of 'water level' is used foi 
modern levelling instrument. A long hollow gla 
cylinder is filled with water with a small air bubbl 
in it, when the cylindrical rod, in kept on lev- 
ground, along the length touching the surface; 
bubble is at centre. The other side of length to 
kept on ground is made flat so that it doesn't roll 

(c) Preparing the ground - 

Ground should be plastered - so that it is no 
destroyed by pressure of walking, wind or rainsi 
A prominently distinct circle was drawn with centr 
as centre of base of sahku. This line also had 1 
permanent or indelible marks by groove gr per*j 
manent marks. Sankaranarayana (869 AD) tells tha| 
lines were drawn with sandal paste. This may 
because sandal was available in his area. 



Three Problems of Daily Motion 387 

Verticality of sanku was tested by means of 
plumb lines (lambaka) on 4 sides. 

It seems that fixed length compasses were 
used for drawing circles. This will be conventient 
for bigger circles and length of radius will not 
change in process of drawing. Hence, the radius 
has always been indicated as fixed. This is not 
necessary for finding perpendicular bisector. 

(2) Cardinal directions : 

Let ENWS (figure 2) be the circle drawn on 
the ground where gnomon is set. Let Wi be the 
point where the shadow enters the circle (in the 
forenoon), and Ei the point where the shadow 
passes out of the circle (in the afternoon). Join Ei 
and Wi. Line EiWi is directed from east to west. 




Figure 2 - Cardinal directions 

Its perpendicular bisector is found by drawing 
two arcs of equal radius greater than 1/2 EiWi. 
This can be any length greater than this, 25 angula 
radius prescribed here meets the condition. A fish 
figure is formed with Ni and Si like mouth and 
tail point of the fish. Since Ei Wi was east west, 
its perpendicular bisector NiSi will be in north 
south direction. NiSi being bisector of chord EiWi, 
it will pass through the centre O and meet the 



388 Siddhdnta Darparj^ 

circle at points at N and S indicating north and 
south. Line EW parallel to E a Wi through centre O 
will mark east and west points E and W on the 
circle. 

Angle points Ai, A 2/ A 3 and A4 between 
cardinal directions can be found by bisecting arcs 
EN, NW, WS and SE. 

As the sun moves along the ecliptic, its 
declination (kranti) changes. By the time shadow 
moves from OWi to OEi, the sun traverses some 
distance of the ecliptic, and its declination changes 
(though very small.) Hence, EW is not he true 
position of east west line. This minute correction 
was described first by Brahmagupta (628 AD), 
Bhaskara II (1150), Srfpati (1039 AD) etc. As the 
correction is very small, this method is good for 
practical purposes. 




Figure 3 



(3) Correction for Kranti change - 

From the east west line ew found as above, 
we make a circle with ew as diameter (Fig 3) Let 
d = correction in ew for change in kranti. 



Three Problems of Daily Motion 389 

O = latitude of the place 

<5 = declination of sun when shadow tip enters 
the circle in forenoon at W 

<5 = Sun's declination when shadow tip leaves 
the circle in afternoon at point e. 

K = chaya karna 

Then d = K (sin a " Sin W 

cos <p 

To apply this correction, a circle with radius 
d is drawn with e as centre which cuts ew circle 
at e' towards north when sun's ayana is towards 
north (as shown in the figure), e' is south from e 
if sun's ayana is towards south. Now e'w is the 
correct east west line. 

This figure is for situation when sun is having 
south kranti with respect to the place, so that 
shadow end is in north part. The south kranti will 
decline in north ward motion of north ayana, hence 
will be at lesser distance in north direction 
compared to w. Thus e' is north from e. 

Derivation of formula : 

Assuming constant declination, w and e points 
have equal shadow lengths, hence their directions 
Ow and Oe are inclined at equal angles from ON 
direction. 

It will be proved that K Sin (5/Cos0 is the agra 
or the distance of shadow from the east west line 
passing through mid day equinox shadow end. 
Hence the change in north south position will be 
difference in the agras at places e and w. 

Hence this was named agrantara correction 
by Caturvedacarya and then accepted by Sripati. 



390 



Siddhanta DarpanaM 



Unfortunately, derivation of this formula is 
not possible without use of spherical trigionometry 1 
in celestial triangles. Three dimensional diagrams! 
are difficult to make on paper, they are approximate J 
indications only. 




Figure 4 (b) Figure 4 (c) 

Fig 4 (a) is yamyottara or meridian circle 
NPZS. (half circle over horizon), SEN is horizon 
showing south (S), East (E) and north (N) points. 
Z is vertical and P is pole of equator EQ. Hence 
ZQES = NP = (p = latitude of the place. In north 
kranti, sun is moving in a diurnal circle R X V 
parallel to eequator towards pole P. In south kranti 
its position will be like R'V. At a position X of 
sun, its kranti is distance Q' from equator measured 
along great circle passig though P. 

Hence PX = 90°- 5. Distance of sun from Z 
is measured along great circle Z X B = ZX = z. 

Figure 4B is the direction circle with sanku at 
O in which WE and NS are direction lines. R is 
the palabha position on equinox mid day. DD' is 
east west line through it. At any instant OS is 



Three Problems of Daily Motion 392 

shadow. Its distance from east west line WE is SM 
called agra jya. Thus agra is the angle a between 
east horizon E and direction X of sun in a circle 
through vertical. Thus a = E X arc or L EOX = 
l_ SOM. In Fig 4(a) it is EB arc. on horizon circle 
(This direction along polar circle is kranti) 
Bhuja of chaya = SM = OP = OS Sin a 

SC = Distance of shadow end from DD', east 
west line on equinox day = Karna vrttagra. 

In APZX, 

Cos (90°- 6) = cos (90°- <p) cos z + sin (9O°-0). 
sin z cos (90° +a) 

where, L PZX = 90° + a 

or, Sin 6 - sin cos Z + cos 0. Sin Z. Sin a. 

K 
Multiply both sides by ^, where K is 

shadow length = V12 2 + s ^ / 12 is sanku and S 

is shadow. Then 

K. sin „_ _, ■ T/ p . «. 

— = K Cos z tan <p + K Sin z. Sin a - 

cos 6 

- - (1) 

But K Cos z = 12, K sin z = S (2) 

from figure 4 (c) 

chaya bhuja b = S Sin a already shown 

Hence b = K sin z. Sin a (3) 

__ Ksin<3 ^ ± _ 

Thus — = 12 tan + b 

cos 6 

But 12 cos (p = palabha = s = equinoctical mid 
day shadow (OR in fig 4b) 



.;■-# 



> A*l 




/Si! 



■^ 



<■&* 



.;$ 



392 Siddhanta Darparj^ 

Ksin<5 

Hence — = s + b (4) 

cos 6 

When Sun is on horizon, ER is agra A in 
4(a). 

In A PRN (ZPNR = 90°) 

Cos (90° - d) = Cos <p Cos (90° - A) 

sin d 

or Sin A = ....(5) 

cos <p 

This agra Jya is in a circle of radius R. 
Reducing it to circle of radius K it is called Karnagra 

Ksind 

a = K Sin A = — 

cos <p 

Thus a = s + b (6) 

In the figure 4(b) Karnagra is difference of s 
and b i.e. Karnagra SC = PR = PO-RO = s - b 

Sum or diffrence depends on opposite or same | 
direction of shadow bhuja and palabha. J 

™ , , K (sin d ' - sin d) \ t , 1 
Thus the formula ■ — is dif- | 

cos <p I 

ference of two shadows in north south directions | 
by which they should be corrected to make its 
ends in true east west direction. 

(4) Alternative methods : 

Vatesvara, Bhaskara I and II, Lalla etc have 
given many other methods also, which deserve to 
be mentioned. 

(a) Mark the points of extremities of two equal 
shadows, one before midday and one after that. 
Line joining them is east west line when . due 
correction is made for change is sun's kranti. 



i 



Three Problems of Daily Motion 393 

This is same as the above method. 

(b) When the sun enters the circle called prime 
vertical shadow of a sanku is exactly in north-south 
direction, i.e. smallest shadow. It will be zero, 
when kranti of sun is same as aksamsa of the 
place, and not useful. 

(c) Bhuja and koti of a shadow (its distance 
from east west or north south line) is calculated. 
Two bamboo strips equal to bhuja and koti are 
taken. Koti strip is laid from centre towards west 
and bhuja strip is laid from shadow and towards 
south, so that their other ends meet. Then koti 
will be in east west direction and bhuja in north 
south. 

(d) Any heavenly body with zero declination, 
rises exactly in east and sets exactly in west. 

(e) The point where star Revati (£ Piscium) 
or sravana (Altair or a - Aquilae) rises is the east 
direction. Or it is that point which is midway 
between the points of rising of citra and svati. 

Only those stars will rise in east which have 
zero kranti Observing citra and svati was used by 
people living in north of,30°N. Sudhakara Dvivedl 
has written in Digmimamsa, that sravana, whose 
celestial latitude is about 30° N cannot rise in the 
east, as it will nevere have 0° kranti (minimum 30 

1 1 

- 23- = 6-° North Kranti). 

(e) The junction of two threads which pass 
through the two fish figures that are constructed 
with the extremities of three shadows (taken two 
at a time) as centre is in the south or north relative 



^94 

Siddhanta Darpanq 

to the foot of the gnomon, according as the sun 
is in the northern or southern hemisphere. 

With the junction of the two threads as centre 
draw a circle passing through extremities of the 
three shadows. The tip of shadow of a gnomon 
does not leave this circle in the same way as a 
lady born in a noble family does not discard the 
customs and traditons of the family. 

Same views had beeen expressed by Lalla 
Sripati and Bhaskara I (629 AD.) But this has been 
rightly criticised by Bhaskara II (1150 AD). As the 
sun is moving on a circle, locus of the line from 
sun to sanku top will be a cone with sanku top 
as apex. Its intersection by horizon plane will be 
always a conic section. The horizon plane is 
inclined at angle (d + <p ) with sun's direction which 
is not 90 °, hence it cannot be a circle. As the 
shadows at sunrise and sunset time are of infinite 
length, they will be in general a hyperbola 
extending up to infinity. When (<5 + cp) = 90° which 
is possible only within polar circle, its locus will 
be circle. 

Siddhanta Darpana has mentioned this view 
in verse 85 of this chapter and has criticised it 
there and in goladhyaya. However, this method 
will be approximately correct if the central position 
of hyperbola i.e. positions near mid day are taken. 

Verse 6 : Relations be- 
tween sanku and Chaya - Add 
the squares of sanku and chaya 
and take the square root of 
sum, which will he chhaya 
karna. Square of sanku (144) is 
substracted from karna square 
and square root of the dif- S Ch5y5 

Figure 5 




OQC 

nree problems of Daily Motion 

( , e nce is chaya which is base or pada. Square root 
^difference of squares of karna and chaya xs bhu,a 
r nku = 12) Line joining ends of bhuja (ianku) 
and koti (chaya or pada) is called karna. 

Note : Relation are obvious from figure 5. 

OV = Sanku = 12 length = Bhuja for the angle 
z of Sun's direction from vertical (Z.SVO - Z) 

OS = chaya = Koti or pada for angle z. 

VS = Line from sanku tip to shadow tip = 
Karna of chaya. 

' OS is in horizontal plane, OV is vertical, hence 

Z.VOS = 90°. 

Thus VS 2 = OS 2 + OV 2 
Verse 7 : Method to find square root. 

Steps - (1) Given a number mark the even 
(sama) places and the odd (visama) places from 
right (unit place) by horizontal and virtical lines. 

-1-1 -1 

Example 11 97 16 

(2) Substract the greatest possible square trom 
the last odd place. 

(3) Always divide the even place by twice the 
square root upto the preceding odd place 

(4) Substract from the odd place (standing on 
the right) the square of the quotient 

(5) Repeat the process as long as there are 
still digits on the right. 

Notes : (1) This method was first given by 
Aryabhata 



■M 



396 



Aryabhata 



-1 -1-1 
119716 (3 



Siddhanta DarpanM 



-A 



2X3 = 


6) 29 (4 
24 




57 
4 2 


2X34 = 


68) 411 (6 
408 


VA = 


36 

-6 2 

X 

346 

New Method 


3 
3 


11 97 16 (346 
9 


64 

4 


297 
256 


686 


4116 
4116 



X X 



This is short version of 

same Aryabhata. 
method. 



(2) Proof of Aryabhata method - 
(1) Put xa = [vTT], xi = 3 

11 - X! 2 = 2 

(ii) Divide 29 by 2xi with quotient x 2/ x 2 = 
29 = 2xix 2 +5 
(iii) 57 - x 2 = 41 

(iv) Divide 411 by 2 (10 x a + x 2 ) = 2 x 34 
411 = 2 (10 Xl + x 2 ) x 3 + 3 
(v) 36-x 3 2 = 



Three Problems of Daily Motion 397 

Thus we have 

11 = xj 2 +2 

29 = 2xix 2 +5 

57 = x 2 2 + 41 

411 = 2x 3 (10xi + x 2 ) + 3 

36 = x 3 2 



Multiply these equations in order by 10 4 , 10 3 , 
10 2 , 10 1 and add. Corresponding terms are can- 
celled, as 

2 x 10 4 = 20 X 10 3 , 5 X 10 3 = 50 X 10 2 , 41X10 2 
= 410 X 10 

we get 

11X10 4 + 9 X 10 3 + 7xl0 2 + 1x10+6 

= xi 2 X 10 4 + 2x! x 2 10 3 + x 2 2 10 2 + 2 xix 3 10 2 
+ 2x 2 x 3 10 + x 3 2 

or 119716 = (xlIO 2 + x 2 .10 + x 3 ) 2 
= (3.10 2 + 4.10 + 6) 2 = (346) 2 
or V119716 '=■■ 346. 

Some times we get smaller number at odd 
place then numbers which will be substracted from 
that. In previous place quotient is reduced by 1. 

1 - 1 - 1 - 1-1 

738915489 (2 

2X2 = 4 )33 (7 Here quotient should be 8 

as 4 X 8 = 32 is less than 
33. But at next stage, we w01 
get 18 - 8 2 = negative Number. 

2X27 = 54 



451 
-l 2 



)33 
28 


(7 


58 

-7 2 


)99 

54 


(1 



sag Siddhdnta Darpana 

V73&915489 

2X271= 542 )4505 (8 = 27183 

4336 This adjustment is to be 

1694 done in short method also. 

-^ 

2X2718=5436)16 308 ( 3 

16308 
09 
3 2 



Verse 8 : Square root of sexagesimal numbers : 

Some numbers are expressed in successive 
divisions of sixty like danda, kala, vikala which 
are called avayava or components. To find the 
square root of such numbers, steps are as follows- 

(1) From the first component i.e. greatest 
division like danda, we substract the greatest 
square number. This gives first part of square root 
in danda (whose square has beeen deducted). 

(2) If the remainder is less than the square 
root danda, then it is multiplied by 3. Then it is 
converted to next lower component (viz kala) and 
number at that position in kala is added. The sum 
is divided by square root in danda multiplied by 
6 and added with 1. Result will be second i.e. kala 
component of square root. 

(3) If 1st remainder is equal or greater than 
danda root then it is multiplied by 2 and 1 is 
added. This is converted to 2nd component kala 
(by multiplying with 60) and number at 2nd 
component is added. Total remaining kalas are 
divided by danda root X 4 + 3. Result will be kala 
component of the square root. 



Three Problems of Daily Motion 399 

Notes : (1) This is a very ingeneous method 
of finding square root, which I have not come 
across in any other text. This method of square 
r0 ot and cube root method in last chapter has not 
come across the modern world. The method is 
explained by examples for both cases. 

Example 1. 

7) 



60° 20' (7 

-7 2 



11 -> 
11x2+1 = 23° 
23x60' +20' = 

7x4+3=31)1400(45' 
124 

160 
155 



This is more than 7 
Thus square root is T 45' 
Test 



(7° 45') 2 = 



/ 



31 



\2 



/ 



961 
16 



= 60 



Which is slightly less than ' 
the square no. 



16 



Example 2 

7) 



50° 20' (7 
-7 2 



1° 

1°X3 = 3° 
3°x60'+20' = 
7X6+1=43) 200 (4.6 

172 



280 
258 



22 



This is less than 7° 
Thus square root is about 

7° 4\6 
Its square is 



/ 



15 
60 



\ 



\ 



'_ 3^ 2 



/ 



40 



/ 



283 x2 



/ 



40 



\ 



= 50° 3' approx. 



/ 



400 Siddhanta Darpar^ 

(2) Justification - This is an approximate 
method, hence an approximate proof or rather 
justification of method is given. 

(i) Suppose A°B' = (a°b') 2 

when A-a 2 > a (Example 1) 

Since A < (a+1) 2 , (a+1) 2 - a 2 > A-a 2 > a 

or 2a+l > A-a 2 > a 

__ . ' (2a + 1) + a 3a + 1 

Hence, A-a z ~ ~ approx = — - — - 

This is multiplied by 2 and 1 is added, then, 
it becomes 

(3a+l) + 1 = (3a+2)° = (3a+2)60' 

Now (a+1) 2 > A > a 2 +a = a (a+1) 

1 3 1 

or A = (a+1) (a+-) = a 2 + - a + - 

= a 2 + 2ja + ^=(a + f) 2 

Hence b = 45' approx (more than half degree) 
B is betwen V to 59' = 30' on average 
Hence remainder is (3 a+2) 60'+30' 
= 180 a + 150 approx 

Dividing by b = 45 , -= = 4fl + 3.3 

approx. 

Hence the remainder is divided by (4a+3) to 
get the value of b. 

(ii)) When A-a 2 < a 

Since A-a 2 > always, on average we can 
take 



Three Problems of Daily Motion 401 

A-a 2 = a/2 

A°B' ~ A+— * ~ a 2 + - + - 

2 ~ 2 2 

, 2 2al ' 1 x 7 

55 ( a + -^r- + 7i ) + 



4 4 2 y 16 

1 . 7_ 
16 



= ^ a + I ) 2 + T^ = ( al / 4 °) 2 = (a° 15') 2 approx 



Remainder is multiplied by 3 and converted 
to kala then added to B - 30' becomes 

| X 3 X 60 + 30 = 90a+30 

On division by 15', range of b it gives 

6a + 2 

Hence it is divided by 6a+l to give approx 
value of b. 

Verse 9 : When in astrology, we calculate 
proportionate life term from value of naksatra/ 
difference of 1 kala will give age difference of 72 
days. Hence component quantity roots should be 
found carefully. This is a rough method involving 
some error. Hence it should be checked by 
squaring. 

Verse 10 : Multiplication of component 
numbers - A multiplication of two quantities with 
3 components each will be in 9 places. First number 
is written at the top with three components at 3 
places. 2nd number with 3 components is written 
below, by its first component we multiply the first 
Hne's components at 3 places. The multiplication 
b y smaller component is written below it, drifted 
1 place towards right. 3rd multiplication by next 



402 Siddhanta Darpt 



tm 



smaller component is shifted 1 more place towards 
right. Thus total is in 5 places. First place fronii 
left is unit (rupa), 2nd place is lipta (1/60 part), 
3rd is vilipta (1/60 lipta) and so on. Only 3 placet 
are taken. Their square root can be found out byg 
method of verse 9. Otherwise, for accurate calculaj 
tion, they will be converted to vikala whose squarei 
root will be in kala. 

Notes : This method is called go-mutrika in 
Indian arithmetic. Like urination by cows at 
separate spots, multiplication is done at different 
lines. Proceduce is as follows - 





a° b' 


c" 


X 








d° e' 


f" 








ad° 


db' 






dc" 






ea' 






eb" 

fa" 


ec"' 

fb r " fc"" 


ad° 


db'+ 






dc"+ 


+ed"' fc"" 




ea' 






eb" + 


+fb"' 


= A 


=B' 






fa" 





= A° =B' =C" 

Only A°B 'C" is kept which is sufficient for 
accuracy. (60 A°+B') X 60 + C" = Vikala 

Vikala = Kala X Kala (e'xb' = eb" vikala as 
above) 

1 1 1 

as — x 



60 60 3600 
Hence square root of vikala will be in kala. 

Verses 11-12 : Setting of sariku 

Circular base of sahku should be plane and 
from top to bottom, face should be plain and 
straight (i.e. smooth conical surface). Height of 



Three Problems of Daily Motion 403 

cone and circumference of base will be equal). 
Shape of sanku may be any type, but l/12th part 
of its height will be called 1 angula. 

For finding out time, our own body also can 
be considered a sanku and the distance of shadow 
is measured from middle point of the feet. 

Convenient sanku is of eye level height made 
of soil or wooden pole. Its centre will be at centre 
of circle. Radius of base is measured already. 
Distance of shadow end is measured from base of 
sanku and radius of base is added to give shadow 
length. 

Verse 13 - The shadow meant here is 
produced by centre of sun. But other parts of sun 
are not dark and they 
also contribute to the 
shadow. Hence the 
length of the shadow 
is increased by 1/211 to 
find the shadow length 
due to sun's centre. 




Notes : In figure 6, shadow of Sanku CP due 
to centre O of sun is CS. Elevation of sun is 
Z.CPS=z. Due to upper most part X of sun, end 
portion SS' is also lighted. Hence, only shadow 
CS' is seen. To find correct shadow, length SS' is 
added to it. Now PS = CP sec z, CS = CP tan z 

S'N is perpendicular on SP. 

Since S'N is very small compared to SP, 



404 


Siddhanta Darpanmi 


S' N sun's radius 
S P sun's distance 


1 ■ : 'M 

= (a known cort*t 

219 m 


stant average value) 




SP 
or S'N = — 





219 

S'S = SN sec z (in right angled triangle S'SN) 
SP sec z CS sec z 



219 



or SS' = 



sin z 
1 



219 



**£m 



-:-*J31 



219 sin z . cos z 



/ 



or ss 



\ 



1 - 



SS' 



\ 



109 sin 2z 
1 



(cs' + ss') 

CS' 
109 sin 2z 



/ 



or 



CS' 109sin2z-l 

Thus the correction will be for less than 1/2 
the distance of SS', because shadow is not dark" 
due to dispersion of light in atmosphere. Logic 
given here is that correection is equal to sun's ; 
radius; distance it is not correct. 

Verses 14-23 : Definitions 

(Text asks to explain the terms through 
spherical model constructed of bamboo to imagine 
the measures correctly. Diagram is a crudle 
substitue, but without it is impossible to describe). 

Sanku is called nara or koti also. 

Chaya is called prabha and bhuja also. 

Square of bhuja and koti added are square of 
karna. 

■ 

This koti, bhuja and karna form fundamental 
triangle. 



Three Problems of Daily Motion 405 

The great circle (straight line for a spherical 
surface) passing through east west points and 
zenith (khasvastika) is called east west circle 
(purvapara vrtta). 

Earth's equator extended into sky is called 
celestial equator (Akasa visuva). Its aksamsa is 
considered zero. Great circle passing through poles 
and east, west points is called samamandala. 

Ahoratra vrtta becomes successively smaller 
as we proceed from equator to meru (pole) 

A sphere of bamboo or wood should be 
formed to show celestial equator, ecliptic, eccentric 
circle of planets and other circles. 

On any day, if the midday shadow of sanku 
is north from sanku, then its difference from 
equinox midday shadow is called agra (more 
correctly karna vrttagra). 

If shadow is south from gnomon (sanku) base, 
then sum of equinox shadow (north for north 
latitude only) and this shadow is called karna 
vrttagra. 

On equinox day sun makes day and night 
equal while on equator (perpendicular to equator 
on that day). Thus the distance of sun on this day 
from svastika of a place is aksamsa or palamsa 
(angular distance from equator) of the place. 

Palamsa is the nati (angular distance from 
zenith or svastika) on equinox midday. Its angular 
height from horizon is unnatamsa equal to 
lambamsa (complementary to aksamsa - distance 
from north pole). 



406 



Siddhanta Darpcti 




12 angula sanku and palabha multiplied by 
radius (3438) and divided by pala karna give 
respectively lambajya and aksajya. 

Notes (1) Figure 7 is as per commentary by 
Pandita Bapudeva Sastri on surya siddhanta. 



■ i! .*;^ 



E 

A 1 


ti^ 


K 


x. P 

\/l \ 


\ / R 

p X. 


\ °\ 1 


X / H 

y f 



,**:# 



XJ 



M 



B 



4 
■■*& 

■'I 

; % 
'"I 



N 



Figure 7 - Definitions in spherical triangles 

ZANB is yamyottara mandala (meridian) pass- 
ing through two poles P, P', and zenith Z. All the 
other circles have been projected on this plane for 
diagram purpose. Samamandala is great circle 
through Z, N and east west points. 

Ksitija (horizon) is circle passing through 
north south east west points. ACB is its diameter 
in the figure which is in north south line. 

Nadi mandala is celestial equator. Its diameter 
is ECF. P and P' are dhruvas (north and south 
poles) of earth. PCP' is a diameter of unmandala 
perpendicular on diameter of nadimandala (or its 
diameter). 

GH is diameter of ahoratra vrtta (diurnal 
circle) of sun (or any planet or star). This meets 






■:J. 



Three Problems of Daily Motion 407 

pCP at L (bisected there) and ksitija at O. Let EM 
be perpendicular to AB. 

Then EZ is aksamsa and CM is its sine or 
aksajya. AE is lambamsa and EM is its sine or 
lambajya. CE is trijya, Thus EMC is a latitude 
triangle with lambajya, aksajya and trijya as its 
sides. It is called W. 

CL is distance between nadi mandala and 
ahoratra vrtta - and is equal to krantijya. L is point 
of intersection of ahoratra vrtta and unmandala 
and LO is perpendicular on line of intersection of 
ahoratra vrtta and Ksitija (this line is perpendicular 
to the plane of paper i.e. diagram). This LO is 
kujya. 

CO lying on ksitija is the distance between 
purvapara and udayasta sutra and is agra (both 
the lines perp. to paper plane). Thus CLO is 
another latitude triangle with sides as krantijya, 
kujya and agra - called X. 

Let the sun be at K. Perpendicular KD to 
ksitija is also called sanku (or mahasanku). DO is 
sankutala and KO, istahrti. OKD is another latitude 
triangle called Y. 

Midday sanku is called madhyahna sanku. 

Suppose sun is at E, the equinoctical point, 
let CR be sanku of 12 angulas. RT is its shadow 
perpendicular to it meeting ECF in T. RT is called 
palabha, and CT is pala karna. CRT is the basic 
latitude triangle called Z. 

Verses 24-27 : Kranti from Palabha 

Now I tell the method of finding current 
declination (angular distance from equator - kranti) 
°f sun forn palabha (midday shadow) 



408 



Siddhanta Darpani 



, . sT.r- 



Midday shadow on north south line is 
multiplied by radius (3438) and divided by karna. 
Arc of this jya is found in kala. This is natamsa 
of sun (distance from kha-svastika = zenith). 

If shadow end is south from the equinox mid- 
day shadow, then sun is having north kranti. 

Then kranti kala of equinox day (aksamsa) is f 
added to natamsa (kala) which gives sun's kranti. 
(for north latitude). Sun's equinox shadow and mid 
day shadow on desired day being in one direction, 
difference of kranti and natamsa is taken. They are 
added when in different direction. 

According to surya siddhanta, palabha (on 
equinox day) is found out from aksajya of the» 
place. Lambajya in found by taking square root of 
difference of squares of trijya (1,18,19,844) and| 
aksajya. | 

Notes 



H 




:'i 



1 



i4i 



N 



Three Problems of Daily Motion 409 

Let HZPN be the observer's yamyottara 
mandate and Z be the zenith. Let EQ be the nadl- 
mandala, HON ksitija and P Dhruva (north). Let 
S be the sun at mid day (In south declination 
towards south point H from Z). S will be towards 
N in north declination. 

ZS is its natamsa or distance from zentih 
(vertical). HS its unnatamsa (elevation from 
horizontal) and SE its kranti (distance from equator 
- shown north here). ZE is aksamsa. 

Draw SA perpendicular to ZO. Then AS is 
natamsajya and OA is unnatamsa jya. 

Produce ZO to cut the circle at Z'. Cut OB 
= 12 angula. Draw BC perpendicular to OZ' 
meeting SO produced at C. Then OB is sahku, BC 
madhyahna chaya (mid day shadow) and OC chaya 
karna. 

Natamsa ^SOZ = ^BOC is given by 

BC Chaya 
Sin ABOC = — -^ 

which is the formula. 

Now when S and E are on same side of Z, 
(as in figure), the shadow BC will be in opposite 
side of both. In this case, SZ = EZ - ES 

Or Natamsa = Aksamsa - Kranti 

When S is on other side of Z i.e. at S', the 
shadow will be in side OZ'H, opposite to equinox 
shadow. Then, 

ES' = EZ + ES' 

Or Kranti = Aksamsa + natamsa 

For same sides it was Aksamsa - natamsa 



410 Siddhdnta Darpana 

Verses 28-32 : Sun from shadow - 

Now I tell the method of finding sun's 
position from shadow. If natamsa and aksamsa are 
in same direction (i.e. shadow on equinox midday 
and desired mid day is in same direction from 
sanku base), then we take the difference of these. 

When they are in different directions, then 
we take the sum. This will give kranti of sun (in 
case of difference, it is in direction of greater 
quantity, for sum, it is direction of either. 

Kranti jya is multiplied by trijya (3438) and 
divided by jya of paramakranti (1370). This will 
give bhuja jya of sun. Its arc is found in kala. If 
sayana sun is in first quadrant, this arc itself is 
position of sayana sun. If it is in 2nd quadrant, it 
is substracted from 6 rasis, in third quadrant added 
to 6 rasis. If sayana sun is in last quadrant, arc is 
subtracted from 12 rasis. 

Ayanamsa is deducted from this value to get 
true sun as measured from mesa 0\ Sphuta or 
true sun is substracted from its mandocca and 
mandaphala correction is done in reverse manner 
for madhyama surya. By repeated procedures, 
madhyama surya will be more accurate. 

Notes : Calculation of sayana sun involves 
two steps (i) Finding kranti of sun as described in 
verse 27. 

(ii) From kranti of sun to its sayana position, 
which has been described in chapter 6 verse 96. 
There the formula has ben used for the reverse 
process, i.e. to find sun's kranti from position of 
sayana sun. 



Three Problems of Daily Motion 411 

Kranti jya x Trijya 
Bhujajya of sun = Parama kranti 

This formula has been proved there. 

Now sayana sun is reduced to true sun by 
reverse process of finding sayana. Earlier ayanamsa 
had bene added (it may be substracted for periods 
before 493 AD or after 2200 AD according to book 
- which is not correct). Hence, it will be substracted 

now. 

Madhyama graha from true graha is again a 
reverse procedure of finding true graha. It has 
been explained in verse 166 of chapter 5. For sun, 
only manda correction is done. 

Verses 33-34 : Shadow from sun's position 
of midday 

Sun's position will give its kranti as explained 
above. Aksamsa of a place is known. If both are 
in different direction, they are added, to give 
natamsa of sun (inclination from vertical). 

If both are in same direction, their difference 
is taken. 

(Here direction of aksamsa is opposite to 
direction of equinox shadow i.e. direction of 
equator from the place). Thus in north hemisphere, 
aksamsa is south). 

Thus we get natamsa at mid day. Its bhujajya 
and kotijya is calculateed 

12 x nat amsa jya 

ch *y a = K^i 

12 x radius (3438) 
chaya karna = ^p 



412 



Siddhanta Darpana 




Figure 9 

Note : This is obvious if we consider figure 
after verse 5 or 13, reproduced here. OV is vertical 
direction at a place where OA is sanku of length 
12. OB is shadow on horizontal plane. Thus ZVAS 
= Z.BAO = natamsa of sun, Z.BOA = 90° 

Now chaya BO = OA tan z = U sm z 



12 x R Sin z 
R Cos z 



Cos z 

12 x natamsa jya 



chaya Karna AB = 
12 X radius 



Kotijya of natamsa 
OA 12 x R 



Cosz R cos z 



Kotijya 

Verses 35-37; Unmandala sanku 

Unmandala is great circle passing through 
east, west points and north and south poles. 
(Defined in verse 23 - figure 7). Its northern part 
lies above horizon in north hemisphere places (like 
India)). Unmandala is horizon of equator, its sanku 
is formed when sun (or a planet) enters unmandala. 
Then perpendicular from it to east west line is 
unmandal sanku. When sun is in north kranti, it 



Three Problems of Daily Motion 



413 



rises earlier than equator, thus at unmandala, it 
has risen at equator horizon and gone above 
horizon at local place. 

Palabha x Kranti jya 
Unmandala sanku = 



yasti = 



Pala karna 
unmandala sanku x trijya 



Carajya 

When sun is north from equator, yasti + U. 
sanku = madhyahna sanku. For sun in south, yasti 
- U. sanku = M sanku. 

NoteS- 




Figure 10 - Unmandala sanku 

ZSZ'N is the meridian of a place of latitude 
3>. S,N. is north south line on horizon. 

ECE' is diurnal (ahoratra) circle's diameter 
when sun in on equator. 

QQ' is diameter of ahoratra when its kranti 
is d . 

P,P' are north and south pole, joining line is 
diameter of the circle passing through east and 
west points on horizon, so perpendicular to plane 
of paper like equator circle. 






41 * Siddhdnta Darpaq* 

PCP' is the north south line of equator and 
unmandala is horizon circle there. C is east point, 
CR is agra. 

Perpendicular from planet at unmandal to 
horizon, is equal to its projection BD in meridian 
plane. Thus BD is unmandala sanku. 

On diurnal circle projection, sun moving from 
Q' above, rises at horizon at point R. At position 
B it is on horizon of equator and rises there. Thus 
sunrise is earlier in north hemisphere when sun 
has north kranti. 

Half of ahoratra vrtta diameter BQ = Dyujya 

Difference between equator and horizon rise 

= BR = Kujya (in kala angles) 

Difference in rising time in asu = Kala for 
equator = CA = Carajya 

EQ = d (Kranti), BC = R sin d= Kranti jya, 
Aksansa <p = arc S'E or PN or angles BCR etc 
marked with Zsign. 

BF and QT are perpendiculars on vertical line 
CZ. Q is mid day time of sun, so TC = madhyahna 
sanku = R sin z 

where z = natamsa QZ = Z.QCZ 

Thus, madhyahna sanku is TF length more 
than BD i.e. unmandala sanku. 

* • 

TF = yasti (or madhya yasti at madhyahna 
time) 

= Height in vertical direction above equator 
rising point. This height at any other position is- 
called ista yasti. 

In latitude ABCD, 



M 



Three Problems of Daily Motion 415 

BD BD 

Sin = ^77 = ~ — : 7 

BC R sin 

or, unmandala sanku BD = R sin 6 Sin $ - - 

-(1) 

(R sin 6) (R sin 0) x . 

or ^ — as stated 

FT = FO + OT = (BO + OQ) Cos<S> 
= BQ Cos $ 

But BQ is at angled from equator 
hence, BQ = R cos d 

Hence yasti FT = R Cos 6 Cos <S> (2) 

yasti = carajya _ 

Unmandala sanku R 

■ * 

by dividing (1) with (2). 

Here, yasti = madhyahna sanku - Unmandala 
sanku (4a) 

when sun kranti is north. In south kranti 
MM', sanku at B' will be in opposite direction. 
Then yasti = madhya sanku + unmandala sanku 
" (4b) 

Value of carajya in (3) has already been proved 
in chapter 6. It is proved as in APCA, BR//CA 

CA BR 
Hence — = — 

CP = R, BR = BC tanO = R sin<5 tanO (from 
diagram) 

BP = R cos d 

Hence carajya CA = R tan d tan O used in 

(3) 



Siddhdnta Darpana 

Verse 38 : Alternative method for madhyahna 
sanku - Madhyahna sanku 

Unmandala sanku X Antya 

Carajya 

Notes (1) Antya = Trijya + cara jya (defined 
later) 

= EC + CA = EA (Fig 10) 

xt BD CT TO + OC 

Now 1ST = ™ (similar triangles) = ~ 

BR QR 5 ; QO + OR 

BD BR CA 

CT ~ QR ~ EA 

or Madhyahna sanku CT 

BD X EA _ Unmandala sanku x antya 
CA Carajya 

when sun is having south kranti, 
Antya = Trijya - Car jya. 

Trijya in asu is half day length at equator, 
carajya is difference in half day length at own 
place. Thus antya in asu is half day length at any 
place. 

(2) Yasti is a stick with length equal to trijya 
= 3438 used to measure vertical height of sun from 
horizon, as ratio of trijya - hence it gives sine 
values. Thus, the height measured from the 
position of equator sunrise is ista yasti. In north 
kranti, at equator rise time, it "is below horizon, 
so its vertical height at equator sunset time can be 
measured, which will be almost equal and opposite. 
For north kranti it can be measured directly. Hence, 
the name yasti has been given. 

Yasti and all sanku measurements are in the 
direction of local vertical i.e. line passing from 



Three Problems of Daily Motion 417 

earth's centre to the surface point. Heights of sun 
along this line from equator rise time will give 
yasti. This gives a measure of equator time i..e 
udayantara correction. 

Verses 39-44 - Agra and Kama Vrttagra - 

Jya of natamsa (R sine of angular distance 
from zenith is called drgjya and its kotijya (R 
cosine) is called sanku jya 

Kranti jya x palakarna 
Madhyahna agra - 12 (4aftku) " " " < A > 

Agra at madhyahna is south or north as sun 
is having north or south kranti. 

Kama Vrttagra 

madhya agra x chaya karna 

= Radius(3438) ( ' 

(Karna Vrttagra is distance of shadow end at 
any time in north direction measured from equinox 
mid day shadow) 

Alternatively, 

Madhyagra - *** *» X .*** (A') 

°^ Lambajya 

Kranti jya x chaya karna mtx 

Kama Vrttagra = - T 7-7-z — (B') 

• & Lambajya 

Say ana sun in six r a si's starting from mesa is 
in north hemisphere and in six rasis from tula is 
in south. 

When sun is in north and karna vrttagra is 
more than palabha (equinox mid day shadow), then 
their difference will be south bhuja or bahu of 
shadow (bahu is length of shadow in north south 
direction). Sun in north and palabha more than 
karna, then their difference will be chaya bhuja in 
north direction. 



418 



Siddhdnta 



When sun is in south, then karna 
and palabha are always added to get chaya 

(These rules have been stated for places 
north hemisphere like India). 

Notes : 





■I 

V _ ■ I | rj 



Figure 11 - Karna Vrttagra 

NZSZ' is meridian, or yamyottara vrtta of a 
place passing through north horizon point N, south 
point S and khasvastika (zenith) Z - i.e. vertically 
up point. 

NES is horizon circle (east half shown) 

P = Pole of equator EQ 

AiR diurnal circle of sun at north kranti 
(declination) 

Ri, R 2/ R3 are its three position. 

Ki, K 2 , K 3 are positions of sun projected on 
equator through polar circles. 

KiRi, = K 2 R 2 = K 3 R3 = Kranti of sun (almost 
equal for a day) 

PRi, PR 2/ PR3 are polar distances of sun. ZEZ' 
is sama mandala through east and west points of 



Three Problems of Daily Motion 419 

horizon, zenith (svastika) points. R 2 is sun's 
position on svastika. 

Polar great circles from Z to positions of sun 
meet equator at Ai, A 2/ E and A 3 . 

Thus natamsa are ZRi, ZR 2/ ZR 3 , angular 
distance from svastika. A 2 Ri, ER 2/ A 3 R 3 are angular 
elevations (unnatamsa) EAi, EA 2 , EA 3 are agras of 
sun. 

Now in spherical triangle PZRi 

Cos PZRi - Cos (FRl) ■" Cos (ZRl) x Cos ( pz > 

sin(ZRi) x sin(PZ) 

PZRi = 90° - agra (a), PZ = 90° - PN 

= 90° - <I>, O = aksamsa 

PRi = PKj-KaR! = 90° -6,6 = Kranti 

ZRI = z natamsa 

TT . Sin 6 - Cosz. Sin <£ 

Hence, sin a = 

sin z . cos <X> 

Sin 6 

- Cot z . tan O (1) 



Sin z . cos O 

_ Palabha ^ 12 

But tan O = — — — , Cot z = — S = shadow 

c . chaya S 

5>m.z = . = — , K = chaya karna 

chaya karna K y 

„ . sin (5 K 12 palabha 

Hence, sin a= — . — - — x 

cosO S S 12 



j. 
S 



K sin 6 

r— - palabha 

cos <I> r 



_ . K sin 6 

or Sana = r~ - palabha (2) 

cos <I> r v ' 



■ - w. 



420 Siddhanta Darpafui 

S sin a = bhuja of chaya measured in north 
south direction from base of sanku. 

Thus, karna vrttagra = bhuja + palabha (By 
definition) 

Ksind ,„, x 

K.V. = — (B') 

cos <I> 

as stated earlier 

Relation (2) holds when sun is having north 
kranti and is north of samamandala. Then bhuja 
is in south direction, which may be taken positive. 

Bhuja (south) = (Karna vrttagra - palabha), 
when in north kranti, sun is south of sama mandala 
angle 'a' is negative (north wards from point E is 
+ve direction). Then 

- Bhuja = K.V. - palabha 

When sun is in south kranti, d will be 
negative, a will be negative so 

- Bhuja = - KV - palabha 

When north direction values are taken 

Bhuja = KV +' Palabha. 

These are the rules for bhuja of chaya. 

Here, madhyahna agra or madhyagra has 
been the name of agrajya at sun rise time which 
may be named A. 

Thus A = R sin ao where a is agra at sunrise 

Then, natamsa Z » 90°, cos Z = and Sin Z 
= 1, equation (1) becoms 

Sin <5 

Sin a = — 

cos O 



Three Problems of Daily Motion 421 

A ■ '.' R sind R x R sin<5 

or, A = R sin a = - — -— - — — — (A') 

cos R cos <X> 

To find (A) and (B) relations, we have 
Palakarna R 

■ 

12 ~ Rcos <f> 

R sin d X Palakarna 
Hence, A = 12 (A) 



From (B'), we have K.V. = 



R sin 6 



cos <I> 



\ 



K 
X R 



madhyagra X chaya karna 

radius 

Verses 45-51 : Relations in sama mahdala- 

When shadow of sanku falls on east west line, 
then shadow, chaya karna and time (indicated by 
nata or unnata amsa of sun) .-•' all are in sama 
mandala i.e east west vertical circle passing through 
zenith. At this point kranti of sun is equal to 
aksamsa of the place. 

When north kranti of sun is more than the 
aksamsa (for north hemisphere) of the place, 
shadow is always south of samamandala. 

Shadow is north of sama mandala when sun's 
north kranti is less than aksamsa of the northern 
place or kranti is south. 

Summary -I'- Shadow on sama mandala - 
then, Kranti = aksamsa 

2. Shadow south ; N. Kranti > aksamsa 
(north) 

3. Shadow north ; N. Kranti < north aksamsa 
or south kranti 



422 Siddhanta Darpann 

(A) Samamandala chaya karna = 
Palabha x lambajya 

jya of north kranti 

Jya of north aksamsa x 12 

Jya of north kranti 

Palabha X dinardha karna 

dinardha vrttagra 

(b) Sama mandala sanku 

Jya of north kranti x palakama 

Palabha 

a 

(c) Drgjya = Vxrijya 2 - Samamandal sanku 2 

drg jy a x 12 

(d) Sama mandala chaya = ' — ; t~t\ 

v ' • ' J Sama mandala sanku 

Trijya x 12 

(E) Sama mandala kama = r - ; — : — \ 

v f • ■ • Samamandala sanku 

(f) Say ana sun bhuja jya 

Sama mandala sanku x Jya of aksamsa 

Jya of parama kranti (1370) 

Notes : (1) When sun is in sama mandala 
(east west circle), sanku, shadow all are in same 
plane. Then ista kala agra a = O. Thus from 
equation (2) after verse 44 - 

Ksiri-d , , , -' 

q _ _ palabha 

cos O r 

palabha x lambajya 

or chaya karna K = — — . . . .. — (A) 

3 • Jya of kranti 

as lambajya = R cos O, Jya of kranti = R Sind 

Kranti is north then, only sun can enter 
samamandala. 

• m 

Palabha X Lambajya = aksajya X 12 
because palabha = 



Three Problems of Daily Motion 423 

12 R sin 
12 tan <D = — - - part 2 of (A) 



K.V. = 



cos 3> 

K sin d 
cos 4> 



^ 11 -1 t^tt Km Sin <5 

For madhyahna, KV m = — (KV m and 

J COS <£ 

Km are value at madhyahna or dinardha) 

Lambajya cos <I> Km 

or . _ = . . = ^T7~ part 3 of (A) 

Kranti jya sin o KVm r 

(2) Sama sanku's and kranti 

Figure 10, in yamyottara plane, indicates 
position O of sun on samamandala ZCB. ZCB, 
unmandala and equator all bisect each other on 
east west points, East point is C here. 

Sama sanku is perpendicular from sun in 
samamandala on horizon. It is equal to perp. from 
O (projection of sun on meridian) to NS, as these 
are parallel projections. 

Thus OC = samasahku 

(height of sun in unmandala) 

ZO is distance from vertex along diameter 
hence angular distance Z is given by 

ZO = R (1 - cos z) = versine Z. 

OC = R cos Z 

In ABCO, ZBOC = <fc (latitude or aksamsa) 

Sin = — — 

OC 

But BC is distance of sun from equator or 
from centre. Angular distance is given by 

R sin d = =BC 



424 



Siddhdnta Darpa^ 



Hence, OC = 



BC 



R sin 6 



Here Sin <I> = 



sin O sin <b 
palabha 



Hence samasanku OC = 



palakarna 

R sin d x palakarna 
Palabha 

R cos z = OC = samasanku 
Drgjya = R sin Z = V R 2 _ R 2 cog 2 z 



- - (B) 



= ^Trijya 2 — samasanku 2 ( c ) 

(3) For other relations, consider figure 12. OZ 
is part of samamandala with centre at P. PG is a 
sanku of length 12 at P. SP and SG are chaya 
karna and chaya of sanku when sun is at O in 
samamandala 

* p 

Natamsa z == arc 
ZO 

= L OPR = L SPG 
= L POC 

OC is samasanku 

OR = PC = R sin Z 
= drgjya 

In similar A s PSG 
and OPC; 




Fig 12 - Samasanku 



SG SP PG 



12 



PC OP OC Samasanku 
Hence, Samamandala chaya SG 
drgjya x 12 



samasanku 



- - - (D) 



Three Problems of Daily Motion 425 



(E) 



Samamandala chaya karna SP 
_ Trijya X 12 
samasnku 

(4) Bhuja of sayana sun - 

R sin<5 

We have R sin (sayana sun) = — — ; , . ... 

v J Sin (parama kranti) 

R sin 6 , ,„* , . 
But samsanku = — ; — — from (B) derivation 

sin O 

or R Sin <5 = Sin O X Samasanku 

Hence R sin (Sayana sun) 

Sin <£ x Samasanku 

Sin (parama kranti) 

Verses 52-62 Kona Sariku 

• * 

From sayana sun bhuja obtained above, we 
can find true and madhya sun as before. Now 
methods for konasanku are explained, which is 
calculated through agra etc. Four points midway 
between east-north, north-west, west-south and 
south-east are called kona (angle directions). There 
are two great circles perpendicular to horizon and 
passing through kona points (one through NE and 
SW points and other through rest two points). But 
they are considered 4, one for each kona point. 

From Surya siddhanta 

Kranti jya x Trijya 

Madhya aera = : — : — t~_ 

J ° Jamba jya 

madhya agra x ista karna 

Karna Vrttaera = „ .. . — LJ 

• ° Trijya 

When the sun enters one of the kona vrttas, 
perpendicular from sun on horizon is called kona 
sanku. Distance of sun from svastika along kona 



426 Siddhanta Darpatj 




n 



vrtta is natamsa and from horizon, it is unnatamsdl. i 
Jya of natamsa (R sin Z) or kotijya of unnatamsa 
[R cos (90°-Z)] is length of kona sanku. 

Shadow of 12 ahgula sanku, then in opposite 
direction of kona is called kona chaya, when in 
north part, kranti of sun is equal to aksamsa, there 
is no shadow in kona directions. 

When midday sun has south nata (altitude), 
then in forenoon, kona sanku is agneya (east south 
angle) and in forenoon, kona sanku is nairtya 
(south west angle). 

When mid day sun has north nata, kona 
sarikus in forenoon and afternoon are called isana 
and vayavya. Now chaya and natamsa can be 
found. 



(A) Karani = 



2 2 2 

12 (Trijya /i — agrajya ) 
_ 

^ + palabha 2 ) 



, Aeraiya x 12 x palabha 

(B) Aksaphala or phala = 6 iJ . ^ 

72 + palabha 2 

(C) Mula = VKarani + aksaphala 2 

(D) Kona sanku = aksaphala ± Mula 

(Sum is done when sun is north of east west 
line samamandala. If sun is south of sama mandala, 
difference is taken) 

Drgjya x 12 



(E) Kona chaya = 



Kona sanku 



^ „ , , Tri Jya x 12 

(F) Kona chaya karna = — ; — : — 

• ' J * Kona sanku 

(G) Drgjya or Kona sahkujya 



Three Problems of Daily Motion 427 

=v Trijya 2 - Konasanku 2 

Results E to G are quoted from surya 
siddhanta 

Notes (1) Equation (1) after verse 44 is 
_ Sin d - Cos z Sin <E> 

Sin a = Sin z Cos <l> 

w here a = agra at any time, d = Kranti of 
sun, z = natamsa, <I> = aksamsa of the place. Sun 
is on konasanku, in forenoon, its agra is 45 ° north 
or 45 "south from east point (according as kranti 
of sun is more than north aksamsa or less). 

1 

Sin a = Sin 45° = yy 

Palabha 
Again we have sin <p = vi?uva j^a or pala karna 

^ = 12 X agrajya 
Visuva karna 

• ■ 

12 
Cos O = 



Visuva karna 

• * 

Then the equation becomes 

Sin a x Sin z. Cos O = Sin 6 - Cos z. Sin <p 

1 12 12 x agra jya 

or 7j Sin z J^^ = pa lakarna 

Palabha 

~ Cos z — — 

palakarna 

(The agra jya on right side is for sunrise time). 

i 
or ^2 sin z x 12 = 12 A " Cos z P- 
(A = Agrajya at sun rise, p = palabha) 



428 



Siddhanta Darpana 



12 : 



or 



Sin 2 z = 12 2 A 2 + p 2 cos 2 z - 2 X 12 

A X p cos z 

But R 2 sin 2 z = R 2 - R 2 Cos 2 Z, Hindu system 
used. 

So, - X 12 2 (R 2 - R 2 Cos 2 z) = 12 2 A 2 + p 2 R 2 Cos 2 z 
- 2 X 12 X A X R cos z x p 



or 



12 2 



R 2 



\ 



- A 2 



= R 2 Cos 2 z [— + p 2 ] 



- 2p X 12XAXR cosZ 

Dividing each side by 12 2 /2 + p 2 

„, , ■ 2 x 12 x A x p „ 

R z cos z Z —z — R cos z 

12 2 



+ P J 



-> R2 

12 2 (y - A 2 ) 



12 : 



= 



+ P : 



Third term is karanT = = N and coefficient of 
R cos Z in second term is called phala = F, then 

R 2 cos 2 z - 2 F, R cos Z - N = O 

or (R cos z - F) 2 = N + F 2 
or R cos z = F ± Vi\f + F 2 

But R cos Z .= R cos (natarhsa) 
= R sin (unnatarhsa) 
= Kona sanku 



So Kona sariku = VKarani + Phala 2 + phaja 



Three Problems of Daily Motion 



429 



Thus F is added for north kranti of sun. 

(2) As in samasanku, it can be proved easily 

Drgjya x 12 



that, Kona chaya = 



Kona sanku 



Trijya x 12 

and Kona chaya karna = — j—r, — 

J ' Kona sanku 

■ 

Drgijya = R cos z = 

OR = PC 

PS = Sanku, GS = 
chaya, 

PG = chaya karna 

OC = Kona Sanku, 
PO = R 

GS PG PS 



PC PO 
gives the result. 



OC 




Figure 12 a 



Verses 63 to 67 : Calculating natamsa - 

Audayika agra 

Jya of sayana sun x Jya of Parama Kranti 

Lambajya 

(Already proved) 

From this formula, when bhujajya of sayana 
sun is 2431 (R sin 45° = 2431) on equinox day, 
then sun rises and sets on kona circles. When sun 
is on equator, it is parallel to east west line at all 
places, hence natamsa of kona circle in forenoon 
is equal to natamsa of kona circle in afternoon. 
(This can happen at 45° north latitude). 

In forenoon, from sayana sun, kranti, chaya 
agra, time etc are found. Sun position at midday 



430 Siddhdnta Darpana 

is approximate and successive approximation is 
needed for kona time. 

* 

Verses 68-71 : Shadow from time and vice 
versa - Now, method is explaine to find shadow 
length, when time is known or vice versa. By this, 
true positions of planet or lagna can be known at 
the time of birth, yajna etc. 

Steps - Natakala is expressed as time or 
equivalent angle, a planet takes to reach mid day 
position in forenoon. In afternoon it is time or 
angle passed from meridian position. 

Nata kala _ Nata kala in kala (N) 

( ' half day " 3 rasi ~ 

This is different from nata arhsa = z which is 
angular distance from vertical zenith, it is more 
than the distance from meridian. 

(2) Utkrama jya vers N = R (1-cos N) is 
found. 

(3) Antya = Trijya ± carajya 

In north hemisphere, when sun is in north 
kranti, sum is used. For south kranti of sun, 
difference is taken 

(4) Unnata jya = Cos N = Antya - Vers N 
Cos N is called ista antya also. 

v , , Cos N x Dyujya 

(5) cheda = T .. . , called ista hrti 

Trijya 






also 



(6) Mahasahku or sanku R cos Z 
cheda x lambajya 

Trijya 

(7) Drgjya = V Tri j y a2 - sanku 2 



Three Problems of Daily Motion 431 

Drgjya x 12 

Triiya x 12 , , ' , 

Chaya karna = — ^~tt already found 

Notes (1) : Formula (6) can be written as 
Sanku 

(Antya - vers N) x Dyujya lambajya 

Trijya Trijya 

(Trijya ± Carajya - vers N) x Dyujya lambajya 

Trijya Trijya 

Trijya - vers N ± carajya . . . 

= — }J — u — X Dyujya X Lambajya 

Trijya 

R cos N ± Carajya 
= ^ X R cos^ X R cos O 

R 2 

or R cos z = (R Cos N ± Carajya) X Cos d X 
Cos O 

d = Kranti, z = natamsa and O = aksamsa 

This formula is to be proved. 

(2) Figure 11 after verse 44 may be referred 
again 

Natakala - Natakala is the time in which sun 
or any other star or planet comes to yamyottara 
(north south vertical circle) in forenoon. In after- 
noon, it is time lapsed since it had come on 
yamyottara. These are called purva and pascima 
nata - incline to east or west. 

Unnata kala is opposite to natakala i.e. time 
taken to rise from horizon in forenoon or the time 
after which the planet will set in west sphere. 

Unnata kala = 1/2 day time - natakala 



432 Siddhdnta Darpanu 

When polar circles to equator are drawn 
through position of sun, the arcs on diurnal circle 
of the planet are proportional to arcs of equator 
which are proportional to rising time in asu when 
arc is in kala or minute. Rotation of earth is along 
equator with fixed speed and time for V rotation 
= 1 asu,. 

Thus in figure 11, natakala at Ri, R2, R3 is 
the time for planet to reach point R of yamyottara. 
Natakala corresponding to points Ri, R2, R3 all east 
from yamyottara are angles ZPRi, ZPR 2 , ZPR3 
which are proportional to arcs QKi, QK2, QK 3 on 
equator. 

Time from E to Q is half day and angle is 

90° = 3 rasi 

Natakala QKi 
Hence, _ .. . — = ~r— f or point Kl; sun at Ri 
half day QE r 

natamsa 

= -^T7r~ - - - Result (1) 
3 rasi 

(3) For sun at Ri, in spherical triangle ZPRi 

Cos(ZRi ) - Cos(PZ) x Cos (PRi ) 
Cos ^ZPRi = Sin (PZ) x Sin (PRi ) 

or cos (nata kala) 

Cos z - cos (90° - <I> ) cos (90° -<3) 

Sin (90° - $) Sin (90° - d) 
cos z — Sin O Sin d 
Cos O Cos d 

Cos z 

= - tan tand (A) 

cos O . cos o 

Now carajya = R tan O tan d (B) 

Adding (A) and (B), 



Three Problems of Daily Motion 433 

Cos z 

R Cos (nata) + carajya = — — — — : — 7 
v ' u Cos 3> cos o 

or Sanku = R cos z 

= [ R cos (nata) + carajya] X Cos <X> cos d 

In Indian system sin and cos are to be 
multiplied by R. Results for obtaining chaya and 
karna have already been proved. 

Verses 72-75 : Time from shadow. 

For this, same formula are used in reverse 
order - 

Chaya X Trijya 
Step (1) Drgjya = ^ ^ 

(2) Maha sanku = VTrijya* - Drgjya 2 

(3) Cheda or ista hrti 

Sanku X Trijya sanku X palakarna 
Lambajya 12 

Cheda X Trijya 

(4) unnatajya cos N = . . ■ ■ / 

(5) Nata Utkrama jya = vers N = Antya-Cos N 

(6) Arc N is found from this. Its value in kala 
is equal to asu of natakala. 

Nata— asu divided by 6 gives nata pala. When 
sun is in forenoon, this is time before noon and 
in afternoon, it is time after noon. 

Notes : Methods can be proved in same way, 
as previous formula. 

Verses 76-77 - When nata utkramajya is less 
than 27 kala, there is a separate method. 



434 



Siddhdnta Darpana 



Natasu = V Antya 2 - Unnatajya 2 x 



- (Trijya + antya) 



Antva 



Note : Utkrama jya is 29 kala for 2nd khanda 
of 7-1/2 \ For smaller values (leess than 7° natamsa) 
this is an approximate method. 

Verse N = (1-cos N) = N 2 /2 for small N 

N 2 
or, — = Antya - unnatajya 

For derivation of this approximate formula 
and to explain the physical significance of terms 
used at each stage, it is necessary to show 
diagrams. 

Natakala has been explained in both circles, 
yamyottara (meridian circle) in Fig 13a and equator 
(visuva) circle in figure 13b. 

In Fig. 13(a), EOE' is diameter of equator, 

Z T 

3< — T^v M 
p 









■I 





Figure 13 (A) 
YSmyottara Vrtta 



Figure 13 (B) 
Visuva Vrtta 



QQ' is diameter of ahoratra vrtta (diurnal circle). 
NS is diameter of horizon in north-south direction. 
In north kranti, sun comes on horizon at K, hence 



Three Problems of Daily Motion 435 

in 1/2 day QK is increased from QR (6 hours) by 
RK. QR = semi diameter of diurnal circle = Dyujya, 
RK = extra length of half day or advance sun rise 
time = Kujya. 

The corresponding lengths on equator circle 
are propositional to time (arc in kala = time in 
asu). Here OE = Radius of celestial circle = 3438' 
kala. OC = carajya. Distance of position X from 
mid day position Q is called nata kala. Correspond- 
ing nata kala on equator is measured by arc EX'. 
EX' = vers N as measured from diameter end E.. 
Length from centre is OX' = Cos N. 

EC = Antya = distance along meridian 
diameter from corresponding positions of sunrise 
and mid-day = EO + OC = Radius + Carajya 

Ista antya for position X of sun is its distance 
along meridian diameter between corresponding 
positions of sunrise and instant position of equator. 

Ista antya = X'C = CE - EX' = Antya - nata 
utkramajya 

QK = Hrti, XK = Ista hrti 

• • * * 

Dyujya = R cos O , where <J> is latitude, 
Corresponding distances on equator and diurnal 
circle are propositional, hence 

!?ta hrti = Hrti _ Kujya Dyujya 

Ista antya antya ~ Carajya " Radius 
= Cos O - - (1) 

Now same positions are represented in Fig 13 
(B) but in equator circle and projections on it. 
Projection of P is at O itself. 

QTQ' = diurnal circle, ET'E = equator circle 
- half portions above horizon EQE' are shown. For 



4^ Siddhanta Darpana 

positon M, when sun has zero kranti, both circles 
are one and nata angle N = ZMOT' = arc T'M. 
Nata utkramajya *■ T'N, Unnatajya or nata kotijya 
= ON and natajya = MN. When N is small, T'M 

(approx) = VqM 2 — ON 2 



or Nata asu = Vjrijya 2 - Unnatajya 2 
In this position antya = Trijya 

Hence the formula, nata = V&ntya 2 - Unnatjya 2 

^ (Trijya + antya) ' 

x — —: = ^Trijya 2 - Unnatajya 2 

This case is proved. 

When sun is having north kranti, horizon 
point on diurnal circle Ki corresponds to horizon 
point C, on equator; so that OKid and OK 2 C 2 are 
in one line. Thus horizons are Ki K K 2 and O C 
C 2 on diurnal and equator circle. 

Here T'C « Antya, TO = Trijya 

At Nata n/ position of sun is at X and X' on 
equator. 

Arc XT' = LX OT' = N 

But sun is seen at X making angle 6 at horizon 
at K. 

T / K = TQ+T approx. as K is almost in 

middle of PC. 

Since angle is small 

XT = X 0, A = antya 



Three Problems of Daily Motion 437 

However, we are measuring angle from C in 
formula 



^antya 2 -unnatajya 2 = A &■. 

A0 R + A 
Hence, N = — x — - — 

__ . — . - — - 1 (R+A) 

or, Hence N = V antya 2 _ unnatajya 2 x 2 — a~ 
(2) Since we are making measurements from 

R+A 

distance T'K = —z—* A cos 6 may be more than 

R. as A > R. Then angle is measured by 
substracting R from A cos 0, as the jya is same in 
next quadrant also. 

Verses 78-80 : Some precautions 

When nata utkrama jya is more than trijya, 
we deduct trijya from it and arc of remaining part 
is taken. It is added to 5400 kala to find nata asu. 

When nata asu is more than 5400 asu, we 
deduct 5400 asu and find jya of remaining arc. 
This added to trijya is nata utkramajya. 

Nata asu multiplied by savana dina (21,659 
asu) and divided by chakra asu (21600) gives 
suksma natasu. 

Notes (1) Calculation for 2nd quadrant is same 
as explained in note (2) after verse 77. 

(2) We are taking a savana dina as 21600 asu 
instead of 21659 asu, hence this proportionate 
correction is done, . 

Verses 81-84 : Sun from agra and sama sanku. 

Now I tell the method to find sayana sun 
from karnagra and samamandala sanku 



438 Siddhanta Darpana 

Karnaera x lambaiya 
«**»*■ Chayakarna ' ' ' ' < A *> 

Kranti jya x Trijya 
Jya (sayana sun) = Jy . rf paramakranti (A 2 ) 

According to the quadrant of sayana sun, 
sayana sphuta sun is found. By deducting 
ayanamsa, sphuta sun is found as before. 

Alternatively, 

Trijya x 12 m 

Samasanku = 7-7: . (Bi) 

Samasafiku chaya karna 

Samasanku x aksajya 

^ ^ Jya of paramakranti 

From (Bi), sun is obtained as before. 

Notes : (1) Formula Ai and A2 have been 
obtained in verse 40-41 or in 53. 

(2) Formula Bi and B2 have been given in 
verse 47-50. 

Verse 85 : According to ancient scientists, 
shadow end of the sanku moves on a circular path 
on a horizontal plane. This is not correct for all 
places and all times. This will be discussed in 
goladhyaya. Now we discuss the method to find 
time in night with help of conjunction of planets 
and stars. 

Note : Locus of shadow has been discussed 
after verse 5. Its formula for radius of circle has 
been given by Vatesvara and Bhaskara II. This is 
correct for only central portion of the hyperbola, 
which is real locus. 

According to Vatesvara, one formula for 

(R + agra) (R - agra) 



diamter of shadow circle is 



Mid day sahkutala 



Three Problems of Daily Motion 



439 




D 



Figure 14 
Diameter of shadow circle 
+ mid day sanku tala. 

In figure 14, circle ENWS with centre O is 
the horizon, with east, north, west and south 
points. A is the point where sun rises, A' is the 
point where sun sets and M is the foot of 
perpendicular on horizon from mid day sun. Then 
circle through A', M and A is locus of shadow, 
approximately for central portion A'MA. 

AB, the distance of A from east west line EW, 
is sun's agra (at rising time). 

MO = Z m , = R sine of sun's zenith distance 
at midday. MF, Distance of M from rising setting 
line AA' is sun's sankutala at mid day. 

MF = MO+OF = MO + BA = Zm + agra 

C is centre of circle A'MA. Let OC = x, Then 

MC 2 = AC 2 (both radius) 

or (MO+OC) 2 = FA 2 + FC 2 

or (Zm -f x) 2 = R 2 - (agra) 2 + (x-agra) 2 

where R is radius of circle E NWS. 

Solving it for x, we get 



440 Siddhanta Darpana 

R 2 - (agra) 2 _ „ 

2x = v ° + agra - Zm 

Zm + agra 

R 2 - (agra) 2 

or 2 (x + Zm) = _ _; 6 _ + Zm 
v ' Zm + agra 

_ (R + agra) (R - agra) + ^ ^^^ 

mid day sankutala 
This gives the diameter, as x + Zm = radius 
of shadow circle. 

Another formula for this diameter is 

(shadow) 2 - (bhuja) 2 + (bhuja mid day shadow) 

bhuja mid day shadow 

This can be proved from same diagram. 

Verses 86-87 : Lapsed or remaining part of 
night is found by observing madhya lagna in sky 
from position of naksatras (position of their stars 
given in a later chapter). Ayanamsa is added to 
madhya lagna. From rising times at equator, lapsed 
part of lagna in the fractional rasi is found. Then 
remaining rising time for sayana ravi at night in 
the part rasi is found. These two are added along 
with rising times of complete rasis between 
dasama lagna and sayana sun. From this sum, half 
solar day is substracted. Remainder is the lapsed 
time in ghati etc of night. Half day added to the 
sum is the ista time from sun rise. 

Similarly, remaining part of 10th lagna rasi, 
lapsed part of sayana sun rasi and complete rasis 
from 10th lagna to sun (rising times) - all added 
and half day of sun deducted gives the remaining 
part of night. 



Three Problems of Daily Motion 441 

Notes : Method of 10th lagna has already 
been explained in chapter 6. 
Verses 88-92 : Rising times of naksatras in Oris- 

sa - 

Method to find lagna has already been 
explained from time of day and night. Now for 
22° Ayanamsa, rising times of different naksatras 
in Orissa are stated, by which true madhya lagna 
can be found in sky. This will be very useful for 
sky watcheers who can satisfy their curiosisty. 

At mid day time, that naksatra is in mid sky 
in which sun is present. 7th rasi of lagna at that 
time is asta lagna (setting rasi). The lapsed times 
of lagna rasis are stated according to the naksatra, 
which has risen in middle sky - starting from 
sravana. 

* 

(22) Sravana - mesa 94 pala (23) Dhanistha - 
mesa 230 pala (24) Satabhisa - Vrsa 280 pala (25) 
Purvabhadrapada - mithuna 24 pala (26) 
Uttarabhadrapada '- mithuna 174 pala (27) Revati - 
Karka 49 pala (1) AsvinI - Karka 187 pala (2) Bharam 
- Karka 256 pala (3) Krttika - sirhha.67 pala (4) 
Rohini - Simha 177 pala (5) Mrgasira - Kanya 2 
pala- "(6) Ardra - Kanya 58 pala (7) Punarvasu - 
Tula 2 pala (8) Pusya - Tula 144 pala (9) Aslesa - 
Tula 184 pala (10) Magna - Vrscika 32 pala (11) 
Purva phalgunl - Vrscika 197 pala (12) Uttara 
phalguni - Vrscika 285 pala (13) Hasta - Dhanu 62 
pala (14) Citra - Dhanu 198 pala (15) Svati - Makara 
12 pala (16) Visakha - Makara 151 pala (17) 
Anuradha - Makara 266 pala (18) Jyestha - Kumbha 
- 67 pala (19) Mula - Kumbha 231 pala (20) 
Purvasadha - Mina 80 pala (21) Uttarasadha - Mina 



442 Siddhdnta Darpana 

1*52 pala. From the difference of rising times of 
these naksatras, time can be found. 

Verses 93-94 - Conclusion - 

Bhaskaracarya II has described many types of 
quantities from bhuja, koti and karna, etc. in 
Triprasnadhikara chapter of his siddhanta siromani 

* 

and has clarified many doubts by questions and 
answers. This already exists in siddhanta siromani 
with his own commentary vasana bhasya. hence I 
am not repeating all due to fear of big size of 
book. 

I have described only those topics in detail, 
which I have verified personally and have separate 
views. This subject can be understood only through 
a good grasp of gola (spherical trigonometry) and 
ganita (mathematical methods). Then derivation of 
formula will not be difficult. Hence I have not 
enlarged the bulk of book by writing proofs. 

Verses 94-95 : Prayer and end - 

May lord Jagannatha fulfil my ambitions who 
is rejoicing with Laksml of unsteady eyes and is* 
residing at Nilacala (Purl) at 276-1/2 yojana north 
from equator i.e. 19°48' N latitude and 200 yojana 
east from Indian prime meridian (passing through 
Ujjain). 

Thus ends the seventh chapter explaining 
three questions (Triprasna) along with views of 
sages; in Siddhanta Darpana written for correspon- 
dance in calculation and observation, and education 
of students, by Sri Candra Sekhara, born in famous 
royal family of Orissa. 



Three Problems of Daily Motion 443 

Appendix to Tripasna dhikara 

(1) (a) Local time, Standard time and true 
time : These three are basis of corrections to planet 
positions, in chapter 2. True time is time cor- 
responding to nata kala; position of sun. Local 
mean time is average time of a locality, assuming 
24 hours in each day. Standard time is local mean 
time of a position taken as standard for a country 
or a time zone. This time differs from Greenwich 
mean time by exact multiples of half hours. Like 
standard time of India is local mean time of place 
82° 30' east of Green- wich i.e. 5-1/2 hours more 
than G.M.T. 

(b) Definitions - Sidereal time - Point of 
equinox from which sayana position of sun is 
measured on kranti vrtta (ecliptic) is moving 
backwards on ecliptic. Position of sun from this 
point along ecliptic is rasi of sayana sun or 
longitude. Position of sun along equator is right 
ascensian. If measured relative to local horizon of 
earth, position of sun along equator is nata kala 
or sidereal time, measured from zenith position of 
sun i.e. 12 hrs noon. Hence right ascension, also 
is written in hours. (It may be called visuva amsa 
or hour angle). 

When motion of equinox is assumed uniform, 
time measured from it, is uniform sidereal time. 
From the true position of equinox, it is called true 
sidreal time. The differnce beetwen them is less 
than 1/10 seconds and normally ignored. 

Sidereal time is west wards, because equinox 
point is moving west wards like sun due to 
eastward daily motion of earth. It is the time in 



444 Siddhanta Darpana 

hours after the instant equinox point has crossed 
the meridian (north south vretical circle of a place). 
Its circle is completed in 24 hours by definition, 
hence 1 hour movement = 15° (=360° -r 24) Position 
of planet in hours of right asencion is 15° per hour 
counted from equinox position along ecliptic. 

c 

ecliptic 
Y 




Equator 

Figure 15 
(c) Mean time - 

Mean sun M is a fictitious point which moves 
along equator with average angular velocity n of 
actual sun. 

Since sun completes one rotation in a sidereal 
year both along ecliptic and along equator, its mean 
speeds are same in both the circles. Mean sun on 
ecliptic is Mi and true sun S. - both coincide at 
perigee or apogee (mandocca). Y is point of 
intersection of ecliptic and equator. 

Y M = right ascension of mean sun 

Y Mi = mean longitude of sun 

Y S = true longitude of sun 

Y M = Y Mi = nt after time t. 

Mean time at any place is called local mean 
time (LMT) Since it will continuously vary at every 
place, local mean time of Greenwich is considered 
standard for the world called Greenwich mean time 
(G MT) 



Three Problems of Daily Motion 445 



G A B Figure 16 

Let A be a place east from G and B another 
place further east. Longitude difference of A and 
B is expressed in hours (1 hour = 15°). 

Let AB = 1 hours = 15 1° 

If S and S' are local sidereal times at A and 
B any instant. 

S' = 1 .+ S 

because Y will cross meridian at B, 1 hours 
before meridian of A its west ward motion. 

Similarly if M and M' are local mean, times 
at A and B at any instant. 

M' = 1 + M 

1 is same in both formulas because hour angle 
and mean sun both increase 360° in 24 hours. 

To avoid inconvenience due to differences in 
the local times of various places in a country, the 
local time of a chosen meridian is regarded as 
standard time. All the places in that country keep 
this time and not the local time. Thus the standard 
time of India is exactly 5-1/2 hours ahead of GMT 
i.e. time of a place 82°30' east of Greenwich. In 
very large countris like Russia or USA, the country 
is divided into zones, each having a differnt 
standard time. For further convenience, the stand- 
ard times of these time zones differ from GMT by 
an integral number of hours or half hours. 

Hour angle measured at Greenwich from 12 
hours noon time is called Greeenwich mean 
astronomical time. (GMAT) and measured from 
to 24 hours. Meantime reckoned from mean mid 



446 Siddhanta Darpana 

night at Greenwich is called Greenwich civil time 
(GCT), GMT or universal time (UT). This also is 
measured from to 24 hours. 

GMT = GMAT + 12 h 

Same is for other places also. 

1. (d) Mean and Sidereal conversion 

In one solar year (tropical), sun crosses 
Y again after one circle. 

It takes K = 365.2422 mean solar days, i.e. K 
revolutions of earth with respect to sun. Hence 
there are K + 1 revolutions of earth with respect 
to Y or any star. Thus 

K+l sidereal days = K mean solar days 

K+l sidreal hours = K mean solar hours etc. 

1 Sidereal days = 1 + — mean solar day 

K. + 1 

= 23 56 4.1 S mean solar units. 

Mean solar day =1+— sidereal days = 24 h 3 m 

A. 

56.5s sidereal hour etc. 

1 (e) Years : Sidereal year is time taken by 
sun for one complete revolution with respect to 
stars on ecliptic which are fixed. 

Tropical year is average interval between two 
successive returns of sun to the first point of Aries 
(Y). As Y moves backwards in about 26,000 years, 
tropical year is slightly shorter. 

(a) Tropical year = 365.2422 mean solar days 
(a) Sidereal year = 365.2564 mean solar days 



Three Problems of Daily Motion 



447 



ecliptic 

e 




(2) (a) Equation of time 
: From the watches we get 
mean solar time only and 
we can get the local mean 
time after longitude correc- 
tion from standard time. To. 
know the true solar time or 
apparent time we have to 
add some correction (+ ve 
or - ve) called equation of 
time. Let 

E = Equation of time. 

a = Right ascension of sun (distance from 
Y along quator = Y D) 

6 = true longitude of sun = Y S along ecliptic. 

1 = mean longitude of sun. 

Then, by definition, 

E = West hour angle of sun - West hour angle 
of the mean sum 

= (S-a) - (S - RA. of mean sun) 

= (S-a) - (S-l) = 1-a 

This can be written as 
■ E = - (a-0) - (0-1) 

Here - (a-0) is called the equation of time due 
to obliquity, because if equator is not oblique a = 
measured along any circle and this term a-0 = 

Similarly, - (6-1) is called the equation of time 
due to eccentricity. 

In the spherical triangle Y SD of figure 16, L 
D = 90°, 

so Cos e = tan a cot 



448 Siddhanta Darpana 

where e is angle beetween equator and ecliptic 

or tan a = cos e tan (0-a+a) 

Expanding this by Taylor's theorem and 
neglecting higher powers of e and 0-a, 

tan a = (1-—) [tan a + (0-a) sec 2 a] 

e 2 
or tan a=tan a + (0-a) Sec 2 a - -y tan a, approx. 

£ 2 £ 

i.e. 0-a = -j Sin a cos a = -j- Sin 2a, 
Since a = 1 nearly, we can write 

£ 2 

- a = ~r sin 21 

which is the required value of - (a-#). Often 
a-0 is called the reduction to the equator, because 
a-0 added to ecliptic co-ordinate reduces it to 
equatorial coordinate. 

Again $ = v+D 

where D is position of perigee and v is true 
amomaly, true position of sun measured from 
perigee. 

1 = m + D 

where m is mean anomaly. 

So, 0-1 = v-m = 2 e sin m, nearly. 

Thus E = 1/4 e 2 Sin 2 1 - 2 e sin (1-D) 

where E, e and e (eccentricity) all are 
measured is radians giving numerical values. 
Expressing it in minutes. 

E = 9 m . 9 sin 21 - 7 m 7 sin (1+78°) 
2. (b) A more accurate value-Put y = tan 2 — 
Then 



Three Problems of Daily Motion 449 

1-y 

COS £ = ~ — — 

1+y 

i-y 

and so tan a = - — - tan 6 

1+y 

e 2ix - 1 
Using exponential formula, tan x = — ^ 

where e = base of natural logarithm, we have 
e 2ia - 1 _ 1-y e^fl - 1 
e 2ia + 1 " 1 + y * ex 2i 6 + 1 

or J* = elid + y - e 2 ^(l + ye- 2 ^) 

1 + ye 2 '" " 1 + ye 2i * 

Taking logarithms this gives 
2 i a = 2i0 + (ye-™ 1/2 yV 4 * + 1/3 y 3 e ^) 

- [ye 2 * - 1/2 y 2 e 4 * + 1/3 y3 e 6i * ) 

= 2i 6 - 2i (y sin 2 0) - 1/2 y 2 Sin40 +1/3 y 3 
Sin 60) 

or 0-a = y sin 2 (9 - 1/2 y 2 sin 4 + | y 3 Sin 
60 (1) 

Again - 1 = v-m 

5 

= 2 e sin M + - e 2 sin 2M + 

4 

= 2e Sin (1-D)) + 5/4 e 2 Sin 2 (1-D) + - - - 

(Introduction to chapter 6) 

Eliminating 0, we get 

E = 1-a == tan 2 | Sin 2 1 - 2 e Sin (1-D) 

+ 4e tan 2 1/2 e Sin (1-D) Cos 21 

- 5/4 e 2 Sin 2 (1-D) - 1/2 tan 4 £ Sin 41 



450 Siddhanta Darpatia 

The equation of time vanishes four times in 
a year. 

E = 9™ 9 Sin 21 - 7™ 7 Sin (1+78°) 

If we draw sine curves f or y == 9 m . 9 Sin 21 
and y =7.7 m sin (1 + 78°) and subs tract one ordinate 
from the other, we get the graph of E, From gr.aph 
is can be seen that it vanishes four times around 
23 march, 22 June, 22 September, 22 December. 

Sin 2 1 attains max numerical values of 1 four 
times in a year and is alternately positive and 
negative at three times. Hence first term twice has 
value + 9.9 minutes and twice - 9.9 minutes 
atternately negative and positive. Thus E is 
alternately positve and negative, because second 
term is smaller numerically. Hence E is zero four 
times a year from theory of equations. 

(3) (a) Parallax : At any instant, the moon 
has slightly different directions as seen from 
different places on the earth. Sun's direction 
changes much less with the change in position of 
the observer, because sun is more distant. In case 
of stars, which are far more distant, the difference 
in their directions as seen from different places of 
the earth is too small to be measured. But seen 
from different places in the earth's orbit, (i.e. at 
different times of the year), the change in the 
direction of the comparatively nearer stars is 
measurable. 

The change in the direction of a celestial body 
as seen from different positions is called parallax. 

For calculation of sun, moon and planets, we 
choose earth's centre as the standard position 
(origin of coordinate axis) from which distances are 



Three Problems of Daily Motion 451 

calculated. Due to observation from surface of 
earth, there is parallax error, called geocentric 
parallax. 

For calculation of star position, sun's centre 
is the standard position and difference in direction 
due to measurement from different positions of 
earth's orbit, is called stellar parallax. 

As geocentric parallax depends upon the 
distance of the observer from earth's ceentre, we 
begin by considering the shape of the earth. 

3 (b) Shape of the earth - Surface of earth 
deteremined by ocean level is called the geoid, 
heights of places above mean sea level being 
negligible. It is an oblate spheroid i.e. rotation of 
ellipse along its minor axis coinciding with polar 
axis of the earth. Semi major axis a of the 
generating ellipse (equatorial radius) is 3963.95 
miles and the semi minor axis b is 3950.01 miles. 
The fraction (a-b)/a is called the compression; 
Eccentricity of this ellipse e = 0.082., compression 
= 1/297. 

Let C be the centre of 
the earth, O the observer 
at any place on its surface, 
OZ the normal at O to the 
surface and OZ' the direc- 
tion which produced back- p . 

ThIn S n? aSS ^ ?° U S h C < angKrveLl 
inen OZ is the direction of 

the astronomical Zenith, OZ' that of the geocentric 

zenith. Angle between these directions ZOZ' is 

called the angle of the vertical indicated by V. 

If <I> and 4>' are the angles made by normals 
NOZ and line COZ' from centre with major axis- 

<& = geographical latitude of O 




452 Siddhdnta Darpana 

<I>' = geocentric latitude of O 

v = O- O' 

If ellipse is referred to C at origin, it is 

x 2 y 2 

— + — = 1 

a 2 b 2 

a 2 y 
Then, Tan O = -^- - - (1) 

b x 

and tan <p'- — tan O 

a 

^ tan O - tan <E> ' 

Thus, tan v = tan (<$-<!>') = 



1 + tan <X> tan <I> ' 
(a 2 - b 2 ^ tan <S> _ (a 2 - b 2 ) sin <S> cos <l> 
a 2 + b 2 tan 2 <I> a 2 cos 2 + b 2 sin 2 <I> 
(a 2 - b 2 ) sin 2 O m sin 2 O 



a 2 + b 2 + (a 2 - b 2 ) cos 2 <f> 1 +m cos 2 * 

a 2 -b 2 
where m = — which is small 

a 2 + b 2 
1 + i tan v 1 + m (cos 20 + t sin 20) 
1 — i tan v 1 + m (cos lsp — i sin Ixp 



2iv I + me 
or, e = 



W 



1 + mc" 2 ^ 

Taking logarithms 

2 iv = log (1 + me 2 ** ) - log (1+me" 2 * ) - 

1 i 

= me 2 ^ - - m 2 e 4i * + — (me -2 ^ m 2 e" 4 ^ +-} 

Hence, v = m sin 2<p - 1/2 m 2 Sin 40 

+ ~ m 3 Sin 6 
3 ^ 

Distance of the observer O from centre C is 
indicated by p 



Three Problems of Daily Motion 453 

V* _ y/b _ 1__ 

a cos f " b sin " ^ (a 2 cos 2 <1> + b 2 sin 2 <&) 

a cos + b sin <p 



So /o 2 = x 2 + y 2 = 



2 2 2 2 

a cos <p + b sin 



a 2 [1 - (2e 2 - e 4 ) Sin 2 <D] 



1 - e 2 sin 2 ^> 



on writing b 2 = a 2 (1-e 2 ) and simplifying 




Figure 18 
Geocentric parallax 

3 (c) Geocentric parallax in zenith distancee 

In figure 18, let C be centre of earth, O is 
observer and M the centre of moon (or sun or 
planet), Let CO = p, CM = r 

If Z' is a point on CO produced, apparent 
zenith distance t! of M is ZZ'OM and true (i.e. 
geocentric) zenith distance zo of M is ZZ'cM. 

Hence z' - zo = parallax in zenith distance 

= ZOMC = p 

From plane triangle OCM 

Sin p = (pk) Sin z' (1) 

Maximum value of parallax p is when z' = 90% 
it is called horizontal parallax p n of M at O. 

Sin p n = pit 

If O is at equator, then the parallax is biggest 
as p has highest value a, equatorial radius. The 
horizontal parallax at equator Po is 



454 Siddhdnta Darpana 

Sin Po = - 

r 

When moon (or the sun) is at its mean 
distance, r G from earth, mean equatorial horizontal 
parallax P is, Sin P = a/r D 

For parallax, earth can be considered almost 
a sphere then, astronomical and geocentric zeniths 
coincide, z'=z, p = constant = a. We take r = r 
approx, then approximate value of parallax is 

p = P sin z - - - (2) 

Since z' > zo, moon, sun or planet is 
distanced away from zenith by distance P sin z 
approx due to geocentric parallax. This is also called 
diurnal parallax as it goes through a complete cycle 
of change through a day. Parallax is maximum 
when moon or sun rise on horizon, reduce to zero, 
when on zenith and again become maximum when 
they set in west horizon. 

3 (d) Distance and size of moon is calculated 
by parallax method only. 

Oi and O2, places on same meridian are 
chosen. Apparent zenith distances of M are 

zi' = LZ\ OiM 

z 2 ' = LZ-l 2 M 




Z2' 



Figure 19 
Moon's Distance 



pi and p 2 are parallax angles Oi MC and O2 
MC, when C is centre of earth 



Three Problems of Daily Motion 455 

If CA is in the plane of equator, 

ZOiC 2 = L Oi CA + Z0 2 CA = Oi + <J> 2 

where Oi and <E>2 are geocentric latitudes oi 
Oi and O2 

Then z'i + z' 2 =^Oi CM + pi+Z0 2 CM + p; 
= 4>i + 4> 2 + pi + p 2 

Thus pi + p 2 = z'i + z' 2 - (Oi + 2 ) = M 1 ! 
Because all values on right side are known, 6 
is known. 

Sin pi = (pi/r) Sin z'i (2) 

Sin p2 = ipiir) Sin z'2 (3) 

Eliminating pi and p2 from the three equa- 
tions, we can know moon's distance r = OM 

It is more convenient to find value of p2 firsl 
and then calculate r. From (1) and (2) 

Sin 6 cos p2 - Cos 9 Sin p2 = (pi/r) Sin zi' 

or Sin & cos p2 = sin z'2 cos + — Sin z' 

r r r 

Eliminating r between this and equation (3), 
we get 

p2 Sin z'2 Sin 6 

tan p 2 = — — ; — : — 

pz Sin z 2 cos + p\ sin z 1 

This gives p2 and then (3) gives r. 




Figure 20 
Moon's diameter 



456 Siddhanta Darpana 

In figure 20, let moon's observed angular 
semi-diameter be S and let its linear radius be R 
miles. If the distancee of moon is r miles as 
determined above, Sin S = R/r from which R can 
be determined. 

In India, parallax in zenith distance is called 
'nati' and parallax in longitude is called 'lambana'. 
Lambana can be measured along equator or along 
ecliptic. Parallax calculation of moon and sun is 
necessary for calculation of solar eclipse. 

3 (e) Lunar parallax along equator and kranti- 

Mean equatorial horizontal parallax of moon is 57' 
and for sun it is 8".80 i.e. 1/388.6 of moon's 
parallax. Hence, accuracy is needed only in 
calculation of moon's parallax. 




Figure 21 
Parallax in natakala and kranti 



D 

11 



In fig. 21, M and M' ar true and apparent 
(due to parallax) positions of Moon (or sun) 

MM' = ^ Sin z' 

r 

where, p = distance of obsrever from centre 
of earth, r = distance of moon from centre of earth 
and, z' = geocentric zenith distance Z'M. 



Three Problems of Daily Motion 457 

Right ascension and kranti of M and M' are 
a„ d and a'(5'. Let H and AH be their hour angles 
(nata kala). 

MD is perpendicular to PM' and L MM'D = 

V 

Small A MM'D can be taken as a plane 
triangle, so 

9 

MD MM sin r\ 



Aa=a -a=-AH=- 

sin PM sin PM 

£ sin z' . sin rj 

r ' cos d 

By sine formula in A Z' PM', 

Sin rj Sin (z' + MM') = cos <I> Sin (H+AH) 

Hence A a = p/r Sin z' Cos O Sin (H+ AH) 
X cosec (z'+MM') / Cos 6 

or A a = p/r cos <J> Sin H sec<5 - - (1) 

neglecting small quantities of second order 

Similarly Ad = 6' - 6 = - M'D =- MM' cos rj 
= pit Sin z' cos r\ 

From cosine formula in A P'ZM 

Sin (z'+MM') cos rj = Cos d' Sin <p' 
~ sin d .cos 4>' cos (H + AH)} 

Substituting this value of cos rj and neglecting 
small quantities of second order 

A 5 = - pit (cos <5 sin <£'- sin d cos <J>' cos H) - (2) 

Regarding earth as a sphere of radius a, we 
can write p and <£' instead of a and <f>. 

Similarly parallax in longitude (along ecliptic) 
and latitude (sara) can be calculated by considering 
P as pole of the ecliptic. Then great circle through 
P and Z' will cut the ecliptic at T called 'tribhona' 
lagna as it is 90° less than the rising point of 



458 Siddhanta Darpana 

ecliptic on horizon or lagna. Hence T = Lagna -90°. 
If t is distance between Z and ecliptic (at T), then 
it is sara of z or declination of T (tribhona). PZ' 
= 90° -t then. In stead of nata kala H we take 
distance of moon from tribhona i.e. v and p is 
latitude in stead of kranti . 

Then (1) becomees, A 1 = lambana 

Al = - p/r cos t Sin v. Sec p (3) 

At eclipse time, P = almost and sec p = 1. 

Equation (2) becomes 

&fi = pit (cos p. Sin t - Sin /S. cos t. cos v) — (4) 

At eclipse time /? = (almost), so cos p = 1, 
Sin p = 

A/3=-p/rsint. (4a) 

(f) Stellar parallax : 

In figure 22, let X' be a star, S the Sun and 
E the earth. Let EX be parallel to SX'. Then EX is 
the true direction of the star, viz its direction as 
seen from the sun and EX' is the apparent 
direction, viz, the direction of X' as seen by the 
observer on the earth. The difference between these 
directions is the angle X'EX which is equal to the 
angle SX'E. 



Figure 22 - Stellar purallax 



Three Problems of Daily Motion 459 

Let L SEX' = 0' L SEX = 

SE = a and SX' = d 

Then from the triangle EX'S in which ZEX'S 
= 0-0', we have 

Sin {0-0') = (a/d) Sin (9 (1) 

Let a/d = Sin f] : then II is called the star's 
parallax (helio centric or annual parallax). Neglect- 
ing second and higher powers of the small 
quantities 0—0' and II, (1) becomes 

0-0' = II sin 

which gives the displacement of the star due 
to parallax. 

EX, EX' and ES are in same plane, so X, XI 
and S are on the same great circle in celestial 
sphere of the observer. Thus the displacement of 
star XX' on sphere = f] Sin XS (S is direction of 
sun on sphere). 

Parallax in longitude and latitude - 

In figure 23, X is 
true position of star 
in celestial sphere, as 
seen from Sun at S. 

X' is its apparent 
position affected by -£35^ 

parallax as seen from 
earth (ceentre of the Figure 23 

sphere). Longitude and latitude of stellar parallax 

MM' is ecliptic and K its pole. M, M' are the 
points on ecliptic at which KX and KX' cut. XD is 
perpendicular from X on KM' 

Let L X' X D = W 
Parallax is II, 




460 Siddhanta Darpana 

X,p andA',/T, are longitude and latitude of X 
and X', then 

A A = X - A = X D Sec£ = XX' cosW seep 

= n sin x s. cos *P sec p 

= II sin MS Sec p 

from the A XMS, in which Z.X = 90° - *V 

i.e. AA = II Sin (0-A) Sec £ (2) 

where is the longitude of the sun. 

Similarly, kp = P' - p = - X'D 

= - XX sin W 

= - II Sin XS. Sin W 

= - II Sin p cos (0 - A ) (3) 

by a PPl)^ n g sme cosmic formula to AKXS 
Parallactic eclipse : If we take X, true position 
of star as origin, XK as the y-axis, where K is pole 
of ecliptic and XD (perpendicular to XK) as x-axis, 
the coordinates (x,y) of the apparent position X' 
of the star are given by 

x = XD = II Sin (0 - A) 

and y = - X'D = - II Sin p. cos (0-A) 

Eliminating 0, we see that locus of (x,y) is 
the ellipse 



x 



2 



y 2 



+ _, * . . = i 



n 2 ' n 2 sin 2 £ 

During the course of a year, the star appears 
to describe this ellipse, which is known as the 
parallactic eellipse. 

If M, M' are taken as positions of X, X 1 on 
equator and T is position of sun on equator, then 
right ascension and declination can be similarly 
calculated. 



Three Problems of Daily Motion 461 

A a = II (cos a cos e Sin 6 - Sin a 6 cos 6) 
Seed 

Ad = II (Cosd Sin e Sin0 - Cos a Sin<5 Cos 6 
- Sin a Sin 6 cos e. sin 0). 

The parallax is used to measure stellar 
distances. Star is seen from the two positions of 
earth six months away (i.e. 180° away in its orbit). 
Direction of a star is seen with respect to a far i.e. 
faint star. Nearest star has parallax of only 0".76 
corresponding to a distance of 

93,000,000 - °- 76 x " ^ 2 55 

60 x 60 x 180 
X 10 13 miles 

This is used to define steller distance in units 
of parsec which is the distance for which stellar 
parallax will be 1". Another unit is light year, 
which is the distance travelled by light in 1 year 
at speed of 1,86,000 miles/sec 

1 par sec = 19 X 10 12 miles.' 
1 light year = 6 X 10 12 

Stellar parallax is not used in siddhanta texts, 
but have been indicated only to show the other 
kind of parallax. Only in goladhyaya it has been 
mentioned (also in discussion of slghra paridhi in 
chapter 51 that stars are 360 times the distance of 
sun. This distance is much more and its parallax 
is no way connected to change of slghra paridhis 
in different quadrants. 

(4) (a) Refraction : The apparent direction of 
any planet or star changes due to bending of rays 
coming from that on earth due to refraction in its 



462 Siddhanta Darpana 

atmosphere. This is called 'Valana' in siddhanta 
astronomy and is calculated empirically. 

Effect of parallax (nati in kranti or lambana 
in longitude) is to shift the planet away from 
zenith. But due to refraction (valana), the planet 
appears higher i.e. closer to zenith. Both are 
maximum at horizon and zero for zenith. 

It is difficult to make exact calculation on the 
basis of refraction rules, even according to modern 
theories of physics. We obtain some formula after 
some simplifying assumptions about variations in 
density and refractive index of different layers of 
atmosphere. In siddhanta books, calculations are 
based on practical observations and the correction 
is assumed to vary according to natajya as in 
parallax. 

According to modern electromagentic theory, 
refraction of light is due to its reduction of speed, 
when it enters a material medium from vacuum. 
Since it is an electromagnetic wave, its speed is 
reduced due to dielectric properties of the medium, 
which has effect like resistance. The reduction in 
speed is more in denser medium. Ratios of speeds 
is called refractive index. 

Speed of light in vaccum 

— *- e = u - Refractive 

Speed in dense medium 

index. 

Since speed of light is maximum in vacuum, 
[i is always greater than 1. When it comes from a 
lighter medium to material of higher deensity, then 
also its speed is reduced 

Speed in medium A u\ L , 

c — — — = constant 

Speed of light in medium B j"2 



Three Problems of Daily Motion 



463 



li, fi\ t and fi 2 are constants for the mediums 
and increase with their density. fi\ and //2 are 
refractive index of mediums A, B. 

N 




Figure 24 - Plane 
refraction 
Due to wave nature of light, a ray AB entering 
a denser medium at B, bends towards normal NNf 
to the boundary surface DE. If its angle of incidence 
with normal is 6 and angle of refraction <b then 
(figure 24) 

sin$ = a 

sin <b ■ fii 

This is a constant depending only on the 
optical properties of the two media. 




Figure 25 Cassini 
refraction 

4. (b) Atmosphere assumed homogenous 
shell- This is called Cassini's hypothesis and is 



464 Siddhdnta Darpana 

simplest assumption. In figure 24, let O be the 
observer on the earth, A a star (or planet) and 
APO a ray which reaches O after refraction at P 
on the upper surface of the atmosphere. Let ju, be 
the refractive index of the atmosphere. Then the 
angles being as marked in the figure. 

Sin 6 = ft Sin 4> (1) 

But from the plane triangle OPC, if radius of earth 
is a, and the height of the atmosphere is h, so 
that CO = a, CP = a+h, we have 

Sin 6 = fi Sin O 
sin t sin O 

— r = - - (2) 

a+h a v ' 

Refraction R = .6 - O - - (3) 

To eliminate 6 and O, from (1) and (3) 

Sin (R+3>) = fi Sin <&► 

or approximately, for small R, Sin R=R, Cos 
R = 1 

R cos O + Sin O = ju Sin O 

Therefore R = (^-1) tan O 

(fi -1) a sin g 

= [(a +h) 2 - a 2 sin 2 £f* by (2) 

{}i -I) sin £ . 

= [cos 2 5 +2 ( h/ a) r a pp roximatel y 

= (a -1) tan £ [1 + <a/ a) sec 2 5] " 1/2 
approximately 

= (a - 1) tan HI - 0/a) sec 2 5] 
= (ju - 1) tan 5 [1 " 0/a) (1 + tan 2 £)] 
which is of the form 
R = A tan £ + B tan 3 £ 



Three Problems of Daily Motion 465 

The simple formula R = K tan £ is true for 
values of £ not exceeding about 45 \ this formula 
is true for values upto 75°. 

4. (c) Concentric layers of varying density : 

This assumption also gives the same formula, by 
an approximate method. 




Figure 26 
Concentric layers of varying density 

Suppose that any layer of the atmosphere is 
bounded by concentric spherical surfaces AB, A'B' 
and that PQR is a portion of a ray of light which 
finally reeaches the observer O on the surface of 
the earth. 

Let C = Centre of earth, CQ = r, CR=r+ Ar 
Then the normal at Q to the surface AB is CQ. 
The angles and refractive indices are as marked in 
the figure. 

From the laws of refraction 

M sin ® = (ju + Ap) sin ^ - .. (1) 

From plane ACRQ 

sin (0 + A <l>) _ sin*P 

r r + Ar 



466 Siddhdnta Darpana 

Eliminating ¥, we get 

fi r sin<D = (a + A/*) (r + Ar) sin (<D + A4>) 
As this relation is true for any two consecutive 
layers, ^ r sin O has the same value for every 
layer. 

But on surface of earth, r = a (radius of earth) 
* = £ (apparent zenith distance) 
fi = ju , (say), depending on density and 
temperature of atmosphere, so 

(jl r sin O = ^0 a sin £ (2) 

Amount of refraction at Q (say AR) = O - W 
so (1) gives. 

fi sin <S> = (w + A//) sin (O - A R) 

= (fi + Ap ) (sin <J> - AR cos 3>) approximately. 

= 11 sin + Aa sin O - A R. fi cos O 

so, AR = (Aw/a) tan4> 

Eliminating O with help of (2), we have 

ddi sin £ A ,.v 

pi (r 2 // 2 - a 2 ^g sin z Q V2 
To solve the differential equation (4), we 
assume 

r 

- = 1 + s 

a 

Where s is small, because the earth's atmos- 
phere extends only to a comparatively small 
distance from earth's surface. Putting this in (4) 
and integrating, we get. 

R = aao sin £ J^ fit' 1 a" 1 (ju - & sin 2 £ + 2s ft 2 ) Vl dfi 



Three Problems of Daily Motion 467 

■d Cf*o -2/2 2-2 ^-V2 

or R = J 1 n (ju -ju sin £) 

Is fi 2 



1 + 



fi 2 - fio 2 sin 2 £ 



d/4 



neglecting higher powers of S. 

It is assumed that z is sufficiently less than 
90° to ensure that the denominator fi 2 - po 1 sin 2 £ 
is not very small 

or R =,io . sin £ £ — j 2 , 2 ^ 

/* (a - fi sin 5) 

. o Sft d/J 

n"o sm£j — . ... (5) 

1 (w z - /^6 sin 2 Q 

To integrate first term, wee put 1/Ja = t then 
it is sin -1 Qio sin £) — £ 

i.e. sin _1 [ (1+x) Sin £] - £ , putting 

To expand the first tern by Maclaurin's 
theorem, 

let f(x) = Sin a [(1+x) Sin £] 

Then f'(x) = , Sm ^ 

Vl - (1 + x) 2 sin 2 5 

Thus f (O) = Sin 1 (Sin £) = £ 

c- 

3nd f (0) = V - sin 2 6 = tan C 

Thus the first term in (5) is equal to x tan £ 
approximately, neglecting higher powers of x. 

Second term in (5), has a small quantity s as 
a factor. So its coefficient is changed slightly. 
Putting // = fio = 1 in it, the term becomes 



" C T7^7 J sd P 



468 Siddhdnta Darpana 

- sing rfl0 

cos 3 C 

Now by Gladstone and Dale's law 

fi - 1 + cp 

where p is the density of the layer with 
refractive index //, and c is a constant. This gives 

dft = cdp 

If p is density of the air at surface. of the 
earth, the second term becomes - 

sin £ 

cos 3 £ 

Integrating by parts, and supposing that s = 
when p - O, this becomes 

i 

COS 3 L, 

The intgrated part vanishes at both limits (p 
= at one limit and s = o at other). The remaining 
integral is equal to mass of a column of air of unit 
cross section, extending from surface of the earth 
to the point P = O. It is, therefore, a constant and 
can be written as B tan £ sec 2 £ , where B is a 
constant. 

Thus R = (a - 1) tan £ + B tan £ (1+tan 2 £) 

which is of the form 

R = A tan£ + B tan 3 £ 

Bradley's formula : He assumed 

r jU n+1 = constant 

Also fir sin <p - constant - from equation (2) 

Therefore, by division 



sin £ rS 



" C TT^T h P ds 



Three Problems of Daily Motion 46$ 



V n 



. - = const. - ^ - (6) 
sin O v ' 

By logarithmic differentiation 

n d0 

- = cos <D . — *- 

From equation (3), — = - tan 4> 

From these two equations 
dR = (1/n) d<& 

Integrating from the surface of the earth 
(where r=a, fi = fi Q and $> = £) to the upper 
boundary of the atmosphere (where fi = l, r=r' and 
O' = O' assumed) 

we get R = 1/n (£-<&')- - - (7) 
From (6), ^ S 1 



i.e. sin 0'= 



sin f sin O 
sin £ 



>o n 



Then (7) becomes, R = -g- sin -1 (sin g) ] 



n n 



This is known as Simpson's formula 
This can be written as 

sin £ 
sTn(e-nR) = M ° n 
or sin £ - sin g - n R) = /«, » - 1 
sin £ + sin (£ + nR) /Lt n + 1 

1 #o n - 1 1 

or tan - nR = ^ tan (£ - - nR) 

Writing 1/2 nR for tan 1/2 nR we get 



4jo Siddhanta Darpana 

R = ^f| tan (^ nR) 
n (jMo + 1) L 

This is Bradley's formula. 

4. (d) Determination of constants - In figure 
27, let Xi and X 2 be true positions of a circumpolar 
star at its upper and lower culminations (positions 
on meridian). Then 




Figure 27 

P Xi = PX 2 = 90° - 6 , 

PZ = 90° - <& 

Therefore, ZXi = (90°- <p ) - (90° -<3) = d - <p 

ZX 2 = 90° - tf>+ 90° - 6 = 180° - <t>- d 

Hence ZXx + ZX 2 = 180° - 2<p (1) 

If the apparent zenith distances at upper and 

lower culminations are £ and ?- then 

ZXi + ZX 2 = Z£ + 2£ (la) 

Z£ = £ -f A tan £ + B tan 3 £ 

Z £' = ?' + A tan?' ' + B tan 3 £ 

Putting this value in (1) we get one equation 

in 5 and - £'. Equation of two more such stars 

will be used to determine A,B and <I> . 

Numerical values of A and B for a pressure 

of 30" of murcury and temperature of 50° F (or 

10 °c) are 58". 294 and - 0". 0668. 



Three Problems of Daily Motion 471 

For values of £ greater than 75 ° , special tables 
are used based on observations. The refraction 
when a body is in the horizon is called the 
horizontal refraction, and its value is about 35'. 

From equation it will be « for £ = 90° as 
tan 90* = oo, hence equation is not correct for such 
values. 

H^— Z 

p 




w 

Y 

Figure 28 

(e) Refraction in visuva amsa and Kranti - In 

figure28, let X be the true position of a star and 
X its apparnt position as affected by refraction. 
Then ZX' X is a great circle and XX' = K tan £ 
where £ is the apparent zenith distance ZX . Let 
the hour angle (natarhsa) and kranti (declination) 
of X be H and d for X' these be H' and d\ 

Join PX; PX' and produce them to meet the 
equator in A and B. Draw X'D perpendicular to 
PM. Then, since XX' is small, the trianlge XX' D 
may be regarded as a plane triangle. 

Now the correction to be added to the 
apparent right ascension a' to obtaint the true right 
ascension (Visuvarhsa) a is a-a'. But 

a-a' = - AB = - X' D sec X'D 

(as X' D is almost equal to arc X' D with 
centre P) * 



472 Siddhanta Darpana 

= - X'X Sin rj Sec <3' 

= - K tan £ sin rj sec 6' 

rj is given in APZX, by sine relation 

sin (90° - O) _ sinH 

sin rj sin £ 

as PZ =90°- <p , ZX = £, so, 

sin ?/ = Sin£ cos Sin H 

Similarly the correction to be applied (added) 
to 6' is <5-<5' But d -6' = - DX = - XX' cos rj 

= - K tan £ . cos?7 

4. (f) Effect of refraction on sun rise and 
sun- set 

Hour angle (natamsa) H of sun's centre when 
rising is (Figure 29) 

CosH = - tan 0tan d (1) 

where <p is latitude of the place and 6 is 
declination (kranti). 

Let H + AH be the natamsa of true sun 
when the apparent sun is rising. At this instant, 
the true sun is really 35' below the horizon, its 
true zenith distance being 90° 35'. Hence, from the 
AP S'Z 

cos (90° 35') = Sin <p Sin 6 + cos <p cos d 
cos(H + AH) 

or, - sin 35' = Sin <p Sin d + cos <f> . cos 6 
(cosH - AH. sin H) 

nearly 

or — Sin 35' = - AH. Sin H. cos <p cos 6 

by (1) 



Three Problems of Daily Motion 473 



N 




Figure 29 
This will give the advance time of sun rise 
H in radian, it will be divided by sin V to get the 
value in asu. 

(1) is obtained from equation for natakala 
cos H. cos O . cos 6 = cos z - sin <p. sin 6 

At sun rise time, z = 90°, cos z = 
Thus apparent day length is increased and if 
sun rise at parama kranti time is measured, it gives 
a higher value of parama kranti. This may be one 
of the reasons for assuming its value as 24° instead 
of 23° 27'. 

4. (g) Shape of sun's disc at sunrise or sunset 

- Lower limb of the sun is at a greater zenith 
distance than the upper. Hence due to refraction, 
the lower limb is raised more than the upper. Thus 
the sun appears flattened. This effect is maximum 
when sun is near the horizon. 

Let S be sun's centre, a its radius and P any 
point on sun's limb, (figure 30) 

Let ZS = z and let PQ be the perpendicular 
from P on ZS. 

On account of refraction, let P be displaced 
to P' and let P'Q' be the perpendicular from P' 



474 



Siddhdnta Darpana 



on ZS. Then, since QP is small, the zenith distances 
of Pand Q are the same. So PQ will be displaced 
to P'Q'. 

Now take SZ as the X axis and perpendicular 
to it through S as the y axis. Then if the coordinate 
of P' are (x,y), we have 

x = SQ' = a cos ^ + QQ' 

= a cosW + K tan (z-a cos *P) 

= a cos + K (tan z-a cos V sec 2 z) (1) 




x-axis 



Y axis 



Figure 30 - Sun disc at rising time 

PP' = K tan z where K = \x - 1 

Its component along PQ is 

K tan z cos ¥ (2) 

But from right angled triangle ZPQ 

cos W = tan PQ. cot z. 

Hence resolved part of refraction in PQ 
dirction is 
K tan PQ = K.PQ, since PQ is small 



Three Problems of Daily Motion 475 

y = P'Q' = PQ - K. PQ 
= (1-K). -PQ = a (1-K) Sin^ - - - (3) 
Eliminating V from (1) and (3) , we see that 
the apparent figure of the sun is the ellipse 

(x - k tan z) 2 y 2 _ j 

(a - ak sec 2 z) 2 + a 2 (1 - k) 2 
Thus sun appears elliptical at sunrise and 
sunset. 




Chapter - 8 

LUNAR ECLIPSE 

Candragrahana Varnana 

Verse 1 : According to views of smartta, vedic, 
purana knowers, there are unlimited good results 
from auspicious works at the time of grahana 
(eclipse) like bath, homa, charities etc. People 
repose faith on tithi calculations after seeing eclipse 
as predicted. Due to this importance, eclipse (solar 
and lunar) is described now. 

Notes : (1) This chapter describes the general 
methods applicable both to solar and lunar eclipse. 
Calculation of solar eclipse needs some special 
methods, which will be discussed in next chapter, 
named surya grahana. 

(2) Auspicious effects of grahana are subject 
of 3rd part of Jyotisa called samhita and need not 
be discussed here. However, calculation of grahana 
is a very complicated process. If such a rare event 
occurs as predicted by calculations, it is an excellant 
proof of correctness of theories and formulas. 

Verses 2-6 : Possibility of eclipse. 

Lunar eclipse - At the ending time of Purnima 
(when moon-sun = 180° exactly), difference of 
moon with rahu and ketu is calculated. When this 
differncee is less than 13°, then lunar eclipse is 
possible. 



Lunar Eclipse 



477 



Solar eclipse - Similarly, at the end of 
amavasya (when moon - sun = 0°), moon and its 
pata (rami or ketu) are calculated. Difference of 
moon from any of the pata being less than 18 *, 
solar eclipse is probable. 

We calculate amanta time (when sun=moon), 
from earth's centre. Pascima nata of candra X 1/3 
is substracted from this time and we again correct 
the true moon at this corrected amanta time. 

Again we calculate, vitribha (tenth lagna) for 
this time. 1/60 of its natajya is added to second 
true moon of this time, when moon and nata are 
in same direction. We substract, when they are in 
different directions. If this is less than 34 then, 
solar eclipse is probable. 

Sometimes, when south nati (in meridian 
circle) is less than 1°30' then solar eclipse is 
probable. When drgvrtta is kranti vrtta, then 
difference of candra and its pata being less than 
7°, solar eclipse is possible. 

Notes : (1) Reason of eclipse - When moon 
passes into the earth's shadow, it fails to receive 
light from sun. This causes an eclipse of moon. 
This can happen only when the sun and moon 
are on opposite sides of earth, i.e. on full moon 
time (Purnima when moon-sun = 180°.) 

M1 




Figure 1 - Lumar eclipse 



478 Siddhanta Darpana 

Let S be the centre of the sun, E of earth. 
The cone touching sun and earth has its vertex at 
V. Then the portion of cone from earth upto V is 
the shadow cone of earth called umbra (bhubha). 
This is completely dark as no light from sun reaches 
in that portion. 

Another cone is formed by tangents in 
transverse direction with vertex in opposite direc- 
tion between earth and sun. The portion of this 
cone after earth and beyond umbra (shadow) is 
partly dark and called penumbra (avatamasa). 

Mi, M4, are points on moon orbit at boundary 
of penumbra, M2, M3 on boundary of umbra. 
Between Mi M2 or M3 M4 portion, brilliance of 
moon is reduced, which are described as colour of 
eclipse but no eclipse is formed. In portion of orbit 
M2 M3 completely within the shadow cone of earth 
(bhubha), there is an eclipse. 

At point 1, moon's disc just starts contact 
with, shadow, this is called first contact or 'sparsa' 
(touch) kala. At point 2 moon's disc just enters 
completely in the shadow called second contact or 
'nirmlana' or 'sammilana' (closing the eyes). When 
complete eclipse is aboui to end i.e. moon's disc 
starts coming out of shadow at point 3, it is called 
third contact or 'Unmllana' (opening the eyes.) 

At point 4, moon completely comes out of 
shadow. It is called fourth contact or moksa kala 
(freedom time.) 

When the moon is not completely eclipsed, 
the times of maximum eclipsed portion correspond 
to 2nd and 3rd contacts. 



Lunar Eclipse 479 



D 



Figure 2 - Solar Eclipre 
(2) Reasons of solar eclipse : An eclipse of 
sun is caused by moon coming in between the 
observer and the sun. If the whole of sun is hidden 
behind the moon, we have a total eclipse. If moon 
covers only part of suns disc, we have a partial 
eclipse. When apparent diameter of moon is smaller 
than sun in a total eclipse, the eclipsed part of 
sun is surrounded by visible circle of sun, it is 
called annular eclipse. These are called 'sarvagrasa, . 
and, 'khanda grasa' or kankana grahana respec- 
tively. This can happen only on amavasya, when 
sun and moon are in same direction. 

In figure 2, if observer is anywhere inside the 
shadow cone of moon AVE, the whole of sun is 
hidden from his view. If he is in the extended 
cone FVG, only the central part of sun is hidden 
by the moon. If the observer is within penumbra 
CAV or VAD (except FVG portion), he will see a 
partial eclipse of sun. It can be seen that at point 
O in extended shadow cone only the inner portion 
BB' of sun is obstructed. In this case, moon is 
smaller, so its shadow cone doesn't reach earth's 
surface. 

In this eclipse also, sparsa or first contact is 
time when eclipse starts. 'Nimilana' is time when 
maximum eclipse starts (or total eclipse) i.e. 2nd 
contact'. UnmHana or 3rd contact is when maximum 



lS 



otb' A 




4S0 Siddhanta Darpana 

or total eclipse is about to reduce. 'Moksa', or 4th 
contact is time when sun is completely visible. 

(3) Why eclipse doesn't occur on every 
purnima or amavasya?- 

The inclina- 
tion of moon's 
orbit to the ecliptic N 
is about 5°. Hence 

the maximum dis- 

. c t Figure 3 - Lunar eclipse not occurring 

tance of moons a 

centre from the ecliptic is 5°. Now the axis of the 

earth's shadow lies in the plane of the ecliptic. 

Moon's diamater is about 1/2° and diameter of 

earth's shadow at distance of moon is about 1-1/2°. 

So moon will touch the shadow, when its centre 

is at a distance from centre of shadow by less than 

1/2 (1/2° + 1-1/2°) = 1° approx. Thus, for most of 

the time, moon passes clear out of the shadow. 

Eclipse is possible only when moon is near 
N, the point of intersection of its orbit with ecliptic. 
The northern point of intersection, from where 
orbit goes north of ecliptic is called rahu and other 
southern pata is caled ketu. Hence, rahu and ketu 
are said to cause eclipse. 

For solar eclipse also, sun and moon should 
be in the plane of ecliptic, so that moon's shadow 
touches the earth. Thus on every amavasya, when 
moon and sun are in same direction from earth, 
solar eclipse doesn't occur. Shadow of moon is 
almost a point when its shadow cone touches the 
earth or it may not touch at all. Thus its radius 
may be taken as zero, at distance of earth (from 
moon). Earth's radius makes an angle of about 1° 



Lunar Eclipse 481 

at moon. Hence as distance between shadow centre 
and earth centre less than 1°, solar eclipse is 
possible. Thus within similar distance of moon from 
its node, solar eclipse happens. 

In solar eclipse, sun is not covered, it is only 
locally obstructed, like obstruction of a cloud. Away 
from shadow cone at a short distance, sun is visible 
because parallax shift of moon is 57' compared to 
8". 8 of sun, which is not obstructed there. 



K 


a Sr 

R-a 




"^N 


^ y 


c 


i 


1 VE# 


M 


* 



V 



Figure 4 - Earth's shadow in moon's orbit 

(4) Size of earth's shadow in moon's orbit. 

S and E are centres of sun and earth 

V is vertex of shadow (umbral) cone of earth.* 

FA is one of generators of cone and v its 
semi vertical angle. 

Let moon touch the umbral cone at N and 
NM be perpendicular to EV. 

Then s, the angle subtended by NM at E, is 
the angular radius of earth's shadow at the distance 
of moon. 

Z.ENA = Pi = horizontal parallax of moon 
approximately as AE is almost perpendicular to 
EM. 

S = sun's angular semidiamter, P = horizontal 
parallax of the sun = a/r. 

a = radius of earth 



482 Siddhanta Darpana 

r = distance of S from E. (sun from earth). 
R = radius of sun. 

Then s = Pi - v from ENV of which Pi is an 
exterior angle. 

= Pi - Z.KES, if KEI IAF 

= Pi-KS/SE nearly as SF is almost perp. to SE 

= P! - (R-a) It 

= P + Pi - S. 

or s = P + Pi - S 

This gives the theoretical value of s, but it is 
found that actual observations give the value 2%. 
larger, because earth's atmosphere absorbs light. 

Angular radius of the penumbra at the 
distance of moon can be shown similarly to be 
P+Pi+S (S is angular semidiameter). 

Approximate value of radius of shadow is 
about 42' after adding 2% for atmosphere. It varies 
with change in distance of sun and moon from 
earth. 

As moon moves 360° in 29-1/2 days with 
respect to sun, i.e. with respect to shadow, it will 
be fully in shadow till it covers (diameter of shadow 
- diameter of moon) = 2X42' - 30' = 54' approx. 
The time in covereing the distance. 

x — x 24 hours - \— hours approx. 



60x360 2 4 

This is the maximum duration of a total lunar 
eclipse. 

(5) Ecliptic limits of Moon— Figure 5 is 
celestial sphere part for observer. N is node of 
moon's orbit. C is centre of earth's shadow on 



M 




Lunar Eclipse 483 

ecliptic. M is centre of moon. Moon's orbit meets 
ecliptic at N which is its node. Angle betwen the 
orbits is i. 

In the diagram 
moon is just touching 
shadow. If C was Q 
when M was at N, 
then NCi is called the 
lunar ecliptic limit. If Figure 5 . Ecliptic Limit of moon 
shadow is nearer then 

moon will definitely pass through the shadow. If 
C is away, moon cannot touch it and there will 
be no eclipse. Since M moves about 13 times faster, 
only moon's motion is being discussed. 

As sun is diametrically opposite to O and 
other node of moon's orbit is opposite to Ni lunar 
ecliptic limit is also the distance of sun's centre 
from nearer node of moon's orbit at the instant, 
moon is crossing the ecliptic. 

Let NC = x when moon is crossing the 
ecliptic. Let n, ni be the angular velocities of the 
sun and the moon (radian per hour) in planes of 
their orbits. Let the time counted* from moon's 
centre passing through node be t hours. Then at 
time t, 

NC = x + nt, NM = n a t 

Taking NCM as a plane triangle, 

CM 2 = (x+nt) 2 + (n^t 2 ) - Irnt (x+nt) cos i- (1) 

CM is a minimum when t is given by 

2 (x+nt) n + 2ni 2 t - 2ni x cos i - 4 nint cos'i 
= 

by differentiating equation (1) with respect to t. 



484 



Siddhanta Darpana 



Substituting the value of t given by this in 
(1) and simplifying, minimum value of CM is 

x ni sin i 

(n 2 + n? - 2nm cosi) V2 " ® 

When moon just grazes the earth's shadow 
in its course along its orbit, the minimum value 
of CM - must be equal to the sum of the radii of 
shadow and moon. Hence (2) is equated with 

| <P+ Pl -S) + Sl 

where S and Si are angular semi diameters of 
sun and moon, P and Pi are equatorial horizontal 
parallax of sun and moon. 

As all the quantities P, Pi, S and Si are 
variable, the lunar ecliptic limit also varies. Its 
greatest value, called the superior ecliptic limit is 
12°.l and the least value, called the inferior ecliptic 
limit is 9°. 5 These limits are for a partial eclipse. 

By equating (2) to the difference of radii of 
the shadow and the moon, we can find limits for 
a total lunar eclipse. 

(6) Commencement of solar eclipse 

When partial eclipse of sun starts, the 
transverse common tanquent BA touching sun and 

M 

B . (W 

C 




Figure 6 - Start of solar Eclipse 



Lunar Eclipse 



485 



moon at B and A respectively just touches earth 
somewhere, say at C. 

Let a, b and R be the linear radii of earth, 
moon and the sun. 

ES = r, EM = n and L MEC > 0, 

ZMES = x 

Then r cos (0 + x) + R = a (1) 

ri Cos 6 = a +b - - - (2) 

Divide (1) by r and (2) by ri and substract. 

We get cos - cos (0+x) =■ — .+■ — + — 



or 2 sin — sin 
2 



/ \ 

x 

6+ 2 



\ 



J 



ri ri 

a_ ^ a R 
ri ri r r 



As x is small and is nearly 90°, this gives, 
approximately, 

x = Pi + Si - P+S 




Figure 7 - solar ecliptic limit 



Solar Eclipstic Limits - The solar ecliptic limit 
is the distance of the sun from the nearer node 
of the moon's orbit, at the moment of new moon, 
if a solar eclipse is just possible on this occasion. 

Let MN be moon's orbit and SN the ecliptic, 
so that N is a node of the moon's orbit and let 
the inclination of the moon's orbit to the ecliptic 
be i. = ZMNS. 



4S6 Siddhanta Darpana 

Let M and S be centres of moon and sun at 
the instant of a new moon occuring when the sun 
is near N. By the definition of a new moon 
(amavasya), longitudes (rasi) of M and S are the 
same. Let P be the latitude of moon when at M. 

Let M', S' be the positions of the moon and 
the sun t hours later, and MSN is taken as a plane 
triangle. 

Let MM' = x 

Then change in moon's longitude in t hours 
is x cos i. 

Then change in the sun's longitude in t hours, 
i.e. SS' is m x cos i, where 

r ate of change of sun's longitude 

m " rate of change of moon's longitude 

Then S'N = SN-SS' = p cot i - mx cosi and 
M' N = P cosec i - x 

If M'S' = D, we have 

D 2 = (P cot i-mx cos i) 2 + (P cosec i -x) 2 

- 2 cos i cos i - mx cos i) (P cosec i-x) - (1) 
Only variable in this is x. Differentiating it 

with respect to x, minimum value of D is given 
by 

(p cot i-mx cos i) (-m cos i) - (p cosec i - x) 

- cos i {-fi cot i - m ft cos i cosec i+2 mx cosi) 

= 

p sin i 

or x = 1 - 2 m cos 2 i + m 2 cos 2 i 
Substitution in (1) shows that the smallest 
value of D is 



Lunar Eclipse 487 

(1 - m) ft cos i 

(1 - 2m cos 2 i + m 2 cos 2 if 1 

When numerical values of m and i are 
substituted, it is seen that the value of this 
expression is very nearly ft cosi, i.e. the value after 
supposing m = (i.e. very small speed of sun). 
Putting, therefore, 

ft cosi = S +Si + Pi - P 

We have the condition that the sun just misses 
being eclipsed. This gives 

ft = (S + Si + Pi - P) Sec i 

as critical value of ft within which ft should 
be for an eclipse to be seen in some part of earth. 

Solar ecliptic limit is the corresponding value 
of 

SN = (S+Si+Pi-P) cosec i 

Its greatest value is 18°. 5 i.e. the superior 
solar ecliptic limit; its least value is 15°. 4 which is 
the inferior ecliptic limit of sun. 

Thus the text mentions only the superior 
ecliptic limits of sun and moon as 18° and 13°. If 
distance of sun from the node is more than this; 
eclipse is impossible, if it is less than the lower 
ecliptic limit 15°. 4 or 9°. 5 eclipse is certain at new 
moon or purnima. If distance of sun is within 
inferior and superior ecliptic limits on new or full 
moon, solar or lunar eclipse may or may not 
happen. Further checking should be done at the 
ending times of purnima or amavasya by lambana 
(parallax) of sun and moon and their true diameters 
and speed. 



488 



Siddhanta Darpana 



(7) The other condition of solar eclipse is for 
a local place. The solar eclipse may happen, but 
it will be visible for only a small belt on earth's 
surface through which moon's shadow cone passes. 

When sun and moon are in same direction 
from earth's centre, the eclipse will be visible from 
a place where difference in parallax of moon and 
sun is less than the sum of their semi diameters 
(= 34') approx. 

O' S 




Figure 8 - parallax in amavasya time 

Figure 8 shows the position of true 
amavasya, when moon M and Sun S are in same 
direction from earth's centre E. When observer is 
at O in this line, i.e. when moon and sun are on 
zenith, then the same position remains. When 
observer is at O', moon is ahead of sun towards 
east by p = L O'SE of parallax. Thus moon will 
be in same direction with sun slightly before true 
position, at true time it goes ahead. Thus for east 
nata amavasya time is before true time and in west 
nata it will be after true amavasya time. 

Surya siddhanta has assumed horizontal paral- 
lax as l/15th of the daily motion of a planet, on 
assumption that the (speed X distance) for the 
planet is constant. Linear speed of every planet is 



Lunar Eclipse 489 

assumed to be same and it comes ot to be (15 X 
radius of earth) as explained in 2nd part of this 
book. For moon this gives correct parallax but gives 
great error for other planets, due to wrong 
assumption of distances. Maximum parallax are 
compared below in vikala. 



Planet 


Bhaskara II 


Modern value 
Minimum Maximum 


Sun 


236.5 


8.7 


'9.0 


Moon 


3162.3 


3186 


3720 


Mars 


125.7 


3.5 


16.9 


Budha 


982.1 


6.4 


14.4 


Guru 


20.0 


1.4 


2.1 


Sukra 


384.5 


5.0 


31.4 


Sani 


8.0 


0.8 


1.0 



Siddhanta darpana has corrected the values 
for sun and moon (through still assuming same 
linear speeds) 
Horizontal parallax for moon = 3388". 22 

Horizontal parallax for Sun 31.63 

Change in sun's parallax is an improvement 
caused due to taking higher value of sun's diameter 
as mentioned in Atharva veda. But still it is about 
3.6 times the true value. 

Changed formula for parallax are 

Daily speed 



Sun max parallax = 



Moon max parallax = 



164 
Daily speed 



14 

Thus the parallax of moon is the distance 
travelled by it in 4 ghatl (60 ghati in a day/15) 



490 Siddhanta Darpana 

according to surya siddhanta and in 4/17 ghati 
according to this book. 

For rough calculation, appendix 3(e) after 
chapter 7 gives the formula (3) as 

Al = - " cos t sin v 
r 

p/r = max. parallax, v is distance from 
'Tribhona' lagna, which is taken as zenith as first 
step. 

Then the correction in ghati is 

4.28 X cos t. X Sin H 

where H = nata kala 

For 45° nata (middle position between 
meridian and west horizon), sin H = 1/ V2, H = 
15/2 = 7.5 ghati. 

This positive correction for pascima nata will 
be 2.5 ghati or 1/3 of nata kala if t = taken 30° 
(nata of tribhona) 

Parallax in sara = p/x sin t. 

Parama nati = 1760 approximately, hence 1/60 
of natajya of vitribha or tribhona lagna is added 
for calculating sara difference of moon. Assuming 
nil sara at eclipse' time, this can be maximum of 
34' for an eclipse to be possible at that place. 

(8) Other condition for solar eclipse - When 
sun is moving on east west vertical line, its kranri 
being equal to latitude of the place, its difference 
with moon when apparent longitudes are equal is 
the north south difference i.e. nati (parallax in sara 
or latitude). When it is less ,than 1/2 (sum of 
diameters of sun and moon) or 1°30' then only 
solar eclipse can happen. 



Lunar Eclipse 491 

When Difference of moon and its pata is less 
then 7° then also solar eclipse can happen. This 
is same as 1°30' difference from ecliptic. 

(9) Greatest and least number of eclipses in a 
year - 

Ecliptic limits are as follows - 

Superior Inferior 
Lunar ecliptic limits 12 \1 9°. 5 

Solar ecliptic limits 18°. 5 15°. 4 

1 Lunar month = 29.5 days 

So, time from full moon to next new moon 
= 14.75 days. 

Node of moon moves backwards, making one * 
revolution in about 19 years. Hence sun makes one 
complete revolution with respect to node in 346.6 
days. Thus, with respect to node, sun moves 

360° x 14.75 , ^o o ■ i. ir i 4-u 
or about 15 .3 in half a lunar month. 

346.6 




Figure 9 - No of eclipse 

(A) Least number of eclipses - 

Figure 9 shows the ecliptic and N, N' are 
nodes of moon's orbit. Let NSi = NS2 = N'S3 = 
N'S 4 . 

= inferior solar ecliptic limit i.e 15°. 4. 



492 Siddhanta Darpana 

Let NM 2 = NM 2 = N'M 3 = N'M4 

= inferior lunar ecliptic limit i.e. 9°. 4 

Inferior limits have been chosen to find the 
most infavourable cases in which no eclipse occurs 
beyond these limits. 

Movement of sun is in direction of arrow. 
Si S2 = 2X15°. 4 i.e. 30° 8 but sun moves with respect 
to N by 2X15°. 3 between two consecutive new 
moons. Thus in travel from Si to S 2 at least one 
solar eclipse is bound to occur, because there will 
be a definite new moon in 30.6 days and sun will 
be within limit of eclipse. 

Suppose now that the eclipse occurs when 
sun is near N, then the sun will be outside NMi 
and NM2 at previous and next full moons (i.e. 
15°. 3 away) while NMi = NM 2 = 9.5 only. Hence, 
there will be no lunar eclipse in previous or coming 
full moons. Thus there will be only one eclipse 
(solar) while sun crosses N. 

Sun will be at N' after about 1/2 X 346.6 = 
173*.. 3 days after it has crossed N. Now 6 lunar 
months occupy 6X29.5 = 177 days. Therefore, about 
4 days after the sun at N'„ there will be a new 
moon. Then sun is only 3.7 X 3607346.6 =3.84 
from N' i.e. will within ecliptic limit of N'S 4 . Thus 
there will be a solar eclipse. The preceding and 
succeding full moon occur out side M 3 M 4 as sun 
moves about 1° in a day. N1M3 = 9°. 5 but N'S 
= 14.75 - 3.84 = 10.91 on previous full moon. In 
next full moon N'S = 14.75 + 3.84 = 18.59. Thus 
there are A no lunar eclipses then. 

If the year began shortly after the sun had 
crossed S 4 , the year will end 365.25-346.6 days after 



Lunar Eclipse 4^3 

the same point relative to nodes, so the year will 
have ended much before sun comes near N again. 

Hence in such circumstances, there will be 
only two eclipses in the year, both solar.. 

(B) Greatest number of eclipses in an year - 

Now in figure 9, let NM a = NM 2 = N'M 3 = 
N'M4 = 12 \1 i.e superior lunar ecliptic limit. 

NSi = NS 2 = N'S 3 = N'S 4 = 18\5, the superior 
solar ecliptic limit. 

Suppose further that there is a new moon as 
soon as the sun enters SiN. There will be a solar 
eclipse then. Time counted from the eclipse is 
indicated by H = half lunar month.. Then we haye 
to examine solar eclipses at new moons at time 0, 
2H, 4H, 6H - - - Similarly lunar eclipses are 

examined on full moons at times H, 3H, 5H 

(i) There is already a solar eclipse at t = 
(ii) At t = H, Sun is at 15°. 3 from Si and 
within MiN at full moon, so there will be a lunar 
eclipse then. 

(iii) At t = 2H, sun has advanced 2X15°. 3 
from Si; so it is within NS 2 and there will be a 
solar eclipse. 

At t = 3H, 4H, HH, the sun will be 

within S 2 and S 3 i.e. outside all the ecliptic limits, 
and there will be no eclipses. 

(iv) At t = 12 H, sun will have advanced 
12X15°. 3 i.e. about 184° from Si i.e. 4° from S 3 . 
So the sun is within S 3 N' and there will be a solar 
eclipse. 



4 9 4 Siddhanta Darpana 

(v) At t = 13H, sun will have advanced 4° + 
15\3 from S 3 , so it is 19°. 3-18°. 5 = 0\8 from N' 
in N'M4 and there will be a lunar eclipse. 

(vi) At t = 14 H, sun will be 0.8 + 15\3 

= 16°. 1 from N' i.e. will within N'S 4 = 18°. 5. So 
there will be a solar eclipse. 

At t = 15 H, 16 H, 23 H, the sun will be 
between S 4 and Si, i.e. outside all ecliptic limits, 
and there will be no eclipses. 

(vii) At t = 24H, sun will have advanced 2X4° 
= 8° from Si, so it is within SiN and there will 
be a solar eclipse. 

(viii) At t = 25 H, the sun will have advanced 
8°+15°.3 from Si, so it is within NM 2 , and there 
will be a lunar eclipse. 

But this eighth eclipse occurs 14.75 X 25 days 
i e. 368.75 days after the first eclipse, i.e. about a 
year and 3-1/2 days after the first. So out of 
8 eclipses, 1st solar or 8 th lunar eclipse has to be 
ommitted in a year. Thus in a year there can be 
maximum of 5 solar+2 lunar or 4 solar + 3 lunar 
eclipses depending upon when the year began. 

(10) Eclipse cycle : In Chaldea, before 400 BC, 
(may be in time of Sargon in 2350 BC approx,) a 
cycle was discovered after which eclipses were 
repeated. This was called Saros cycle of 18 years 
10.5 days or 223 synodic lunar months. 
223 synodic months = 6585.321 days 
242 dracontic months = 6585.357 days 
=19 X 346.62005 days (Dracontic year) 
Draconitic year is revolution of sun with 
respect to lunar node and draconitic month is 



Lunar Eclipse 435 

revolution of sun with respect to its node. Nodes 
of moon were called Dragons. 

Visvamitra had mentioned half cycle in 
Rkveda of 3339 tithis = 111 synodic months + 9 
tithis. 

Example of the cycle for least no. of eclipses 
in given below - (No lunar eclipse + 2 solar eclipses) 



Years 
7915 
1933 
_1951 
"1922 
1940 
2958 
"1926 
1944 
1962 



Feb. 14 
Feb. 24 
March 7 
March 28 
April 7 
April 19 
Jan. 14 
Jan. 25 
Feb. 5 



Dates of solar eclipse 
Aug. 10 



Ann. 



Total 



Aug. 21 
Sept. 1 
Sept. 21 
Oct. 1 
Oct. 12 
July 9" 
July 20 



All annulus 



Total 



Annular 



July 31 

Cycle of years of maximum eclipse 

Years Lunar Eclipse Solar Eclipses 

1917 Jan 8, July 4, Dec 28 Jan 23, Jun 19, July 19, Dec 14 
1935 Jan 19, July 16, (Jan 8) Feb 3, June 30, July 30, Dec 25 
1953 Jan 29, July 26, (Jan 19) Feb 14, July 11, Aug 4, gan5), 

next year next year 

Total Total Total Part Part Part Annular 

Actual determination of eclipse, is by calculat- 
ing the extent of eclipse according to true speeds 
and sara as explained later. 

Verse 6 : This book has used different 
methods for lambana correction for sphuta amanta 
(new moon day), true positions of sun and moon, 
dimensions of sun, moon and shadow, grasa 
(covered) amount of moon, sthiti (total eclipse time) 



496 Siddhanta Darpana 

vimarda (total time of complete or maximum 
eclipse), real true lambana, sphuta nati, digvalana 
and parilekha etc. This will be useful, so learned 
men should not think it to be incorrect. 

Notes : Many of the methods have not been 
approved by earlier siddhanta works, but these 
methods give more correct results. Hence this 
needs to be accepted more eagerly, instead of 
rejecting it. 

His methods for different methods of moon's 
motion has already been mentioned in chapter 6. 
Correction of moon's and sun's motion is also due 
to his revised values of manda paridhis which 
change continuously in quadrants. For moon, only 
one maximum value has been indicated and its 
ratio with least value should be increased. Earlier, 
either the manda paridhi was fixed or a fixed 
difference of 40' was kept at the end of odd and 
even quadrants. 

Lambana and nati formula have been cor- 
rected due to changed formula of maximum nati. 
For moon this is taken as l/14th of daily motion 
instead of general formula of l/15th of daily motion 
for all planets. For sun it is entirely changed to 
1/164 of daily motion, which has no parallel in 
earlier texts. The correct variation of nati and 
lambana has been calculated instead of rough linear 
method. 

Value of sun's diameter and consequently its 
distance from earth has been increased about 11 
times the traditional value of 6,500 yojanas to 72,000 
yojanas as mentioned in Atharvaveda. This has led 
to other changes in constants and methods. These 
corrections have been in right direction and more 
accurate. 



Lunar Eclipse 497 

Verses 7-8 - Correct time of parvanta - 

On amanta or purnanta day (moon-sun = 0° 
or 180°), sun and moon will be made sphuta (at 
sunrise or midnight time. For parva ending, only 
mandaphala correction is needed in moon. On 
amavasya day, difference of moon and sun is taken, 
on purnima, it is moon - (Sin + 180°). Difference 
rasi etc is converted to para (1/60 vikala) and is 
divided by difference of sphuta gati of moon and 
sun in kala. Result will be in vighati (pala). 

This time in pala etc is added to parvanta 
time i.e. to sunrise time for which calculations had 
been done, if moon is less than sun (or sun+180° 
on purnima). If moon is more, it will be sub- 
stracted. Then we get the correct time (after or 
before sunrise for ending time of parva (purnima 
or amavasya). For this time, we again calculate 
sphuta moon and sun and from these values, 
correct parvanta time is calculated. After repeated 
applications of the method we get correct parvanta 
(for centre of earth). After that, other corrections 
for eclipse are made (like lambana or nati) for 
observation from surface of earth. 

Notes : As first approximation speed at 
parvanta is assumed to be same as at sunrise time 
and accordingly correct time is calculated. Our aim 
is to find the time when moon-sun or moon- 
(sun+180°) is zero. If moon is less than this value, 
it will cover up the distance due to higher speed. 
The difference is in para (1/60 vikala), speed diff. 
is in kala/day. 

Hence result time = _ f — 

kala/day 



498 Siddhanta Darpana 

Kala x 60 X 60 

_ X ^ a y - p a J a e fc 

kala J r 

After finding approximate parvanta time, we 
get better approximation of sun and moon position 
(their difference and their speeds. Then we get 
more correct value of parvanta. 

Vrses 9-11 - Samaparva Kala - When for sun, 
the mandaphala, gati phala and udayantara phala 
- all three are positive or negative, we further 
correct the samaparva kala i.e. middle point of 
eclipse is slighdy different from true parvanta 
above. Steps are as follows - 

(1) (Udayantara + bhujantara of moon) + (gati 
phala of sun) = S 

(2) S X mandaphala of moon = P 

(3) On purnima, X 

P 

moon diameter ( 444 yojana ) 

P 

On amavasya, X = Sun ^^^ (72/000 yo jana) 

X in vikala _ . , , . . 

(4) : : = L in danda pala etc. 

' Moon gati - Sun gati 

(5) When mandaphala, gati phala and udayan- 
tara phala all are positive, 

Sama Parvakala = Parvakala - L 

When the three above are negative 

Samaparvakala = Parvakala + L 

(6) For this difference of time we further 
correct the positions of sun and moon at parvants. 



lunar Eclipse 4 " 




Figure 10 



Notes : (1) Before analysing the formula we 
should analyse the reasons as to why closest contact 
will not be at amanta or purnimanta time. 

E is shadow of earth centre moving on ecliptic 
for lunar eclipse. For solar eclipse it is disc of sun. 
M is centre of moon moving on its orbit in direction 
MPN. 

At point EM, when EM is perp. to NE, 
ecliptic, longitudes of E and M are same which is 
ending time of amavasya or purnima as calculated 
earlier. However, closest approach is at P when 
EP is perpendicular at P. Thus the real mid point 
of eclipse will be after purnimanta time. When RM 
is after crossing N, then it is before parvanta time. 
This difference is due to inclination of moon's orbit 
with ecliptic and difference PM is given by 
udayantara phala of moon in latitude along ME 
direction and bhujantara phala in EN direction. 

Another reason of difference is due to 
different speeds at points of contact before P and 
after P. Due to that the mid point will be shifted 
from P in ratio of speed difference given by 
mandaphala of moon. 

Udayantara and bhujantara phala of moon are 
almost for same time difference as sun, as moon 



500 Siddhdnta Darpana 

and sun or earth's shadow are in same position 
almost. The result for shadow at 180° from sun is 
same. If speed of moon is increasing, the time in 
covering contact distance towards N after P will 
be less and mid point will be towards opposite 
direction i.e. deducted. 

Similarly for other results positive, the time 
is to be deducted. If mandaphala is + ve, gati 
phala of sun is negative, hence relative motion of 
moon will be positive and it is to be added. 

Thus the difference due to latitude difference 
is (udayantara + bhujantara) of moon + gatiphala 
of sun. This will be increased in the ratio of 
mandaphala of moon. For outer contact, moon will 
cross its (own diameter + shadow portion). For 
inner contact (maximum) it will cover (shadow - its 
own diameter). Hence the product is to be divided 
by angular diameter of moon. In solar eclipse, it 
is almost equal to diameter of sun. 

There appears some error in text. All the 
quantities are in angular measure, which cannot 
be divided by yojanas, it should be angular 
diameter. 

When all the three factors causing error are 
of one sign, correction is proposed, otherwise they 
almost cancel each other. 

Qualiative discussion will be done at the time 
of calculating duration of eclipse. 

Verses 12-15 : Diameters and distances of sun 
and moon- 
In Atharvaveda, while explaining the meaning 
of 'Aum', diameter of solar disc has been stated 
to be 72,000 yojanas. Based on this statement, I 



Lunar Eclipse 



501 



have corrected the disc sizes of planets, their orbits 
etc. through observation and calculation. 

Diameter of moon and earth are 1/162 parts 
and 1/45 parts of sun's diameter. Earlier 
astronomers also have stated the diameter of earth 
as 1600 yojana (value obtained here). The values 
in yojana and angle are stated as follows - 

Diameter in yojana Angular diameter mean 
Sun 72,000 32/32/6 kala 

Moon 444 31/20 kala 

Earth 1600 — 

72,000 yojana 



Mean sun diameter = 



Mean moon diameter = 



2213 
444 X 6 
85 



Mean distance of sun from earth = 76,08,294 
yojana 

moon = 48,705 yojana. 

From this true distance, manda spasta karna 
also can be calculated. 

As in case of moon's angular diameter, earth's 
shadow's angular diameter also can be known in 
moon's orbit (approx by multiplying with 6/85). 

Notes (1) Comparative sizes of planets 

Aryabhata I, 
Lalla, 
Bhaskara I 



Sun's diameter 
Sun's distance 



4410 



Surya 


Modern values 


siddhanta, 


in yojana 


Siddhanta 


= 5 miles 


siromani 

• 




6500 (6522) 


1,73,156 


6,89,378 


1,85,80,000 



502 Siddhanta Darpana 

Moon's diameter 315 480 430 

Moon's distance 34,377 51,566 47,500 

Earth diameter 1050 1600 1586 

Diameter of earth is a measure of yojana as 
its astronomical definition. Hence; it is seen that 
diameter and distance of moon are almost accurate 
in surya siddhanta or others, but sun's diameter 
is taken only 4 times the earth or 14 times moon 
by Aryabhata (13.37 times by Bhaskara II or surya 
siddhanta). Its real value is 109.18 times earth's 
diameter or 402 times moon's diameter. 

However angular diameters were almost cor- 
rect. 

Bhskara II Surya Siddhanta Modern 

Siddhanta Darpana Values 

Moon 32/1 32/0 31/20 31/7 

Sun 32/31/33 32/20 32/32/6 32/4 

Angular diameters and their ratios are almost 
correct. Moon's angular diameter can be direcdy 
observed, but it is difficult to see sun directly. Still 
it can be seen through reflection etc, and due to 
frequent annular eclipses its mean diameter has 
been taken slightly more than moon. 

Linear diameter is calculated by formula 
(angular diameter x distance), when angle is in 
radians. This rato is almost 1/108 , this 108 is an 
important number for no. of beads in a prayer 
garland, no. of salutes to guru, astottari system of 
dasa in astrology etc. Moon's distance could be 
correctly estimated with direct parallax, but direct 
measurement of sun's distance cannot be done. 

The accurate looking figures of distances of 
sun and moon are derived from round figures of 



Lunar Eclipse 503 

their circumference of their orbits after division by 
2jt = 2 X 355/113 almost. On moon's orbit V has 
been assumeed eequal to 15 yojana by Surya 
siddhasnta and 10 yojanas by Aryabhata. Linear 
velocity of planet = (angular velocity X distance) 
has been assumed constant. Actually areal velocity 
= angular velocity X (distance) 2 is constant accord- 
ing to Kepler's laws for elliptical orbits. Thus all 
planets are assumed to cover equal distance in 
equal time and total distance covered by them in 
a kalpa is equal to orbit or circumference of sky. 

Accordingly, orbit of stars has been assumed 
60 times orbit of sun. Candrasekhara must have 
seen distances of farther planets like pluto 40. times 
sun's orbit. Hence he increased it to another round 
figure 360 and explained difference of 1° sighra 
paridhi difference according to this, which is not 
correct. 

Similarly, he must have come across much 
larger figure of sun's distance and verified it 
according to parallax in solar eclipse. But he could 
increase it only 150 times moon's distnace instead 
of 400 times as he got diameters of 72,000 yojana 
from Atharva-veda. Earlier astronomers also must 
have obsereved it, but they didn't try to change 
it drastically, as the angular measure is sufficient 
for prediction of eclipse. Traditional value of surya 
siddhanta appears to be obstruction. 

Siddhanta darpana has assumed value of 
yojana in Atharvaveda as his own yojana which is 
incorrect as Aryabhata etc. had assumed yojana of 
about 8 miles; compared to 5 miles yojana of surya 
siddhanta. Of course, he has compared 1600 yojana 



504 Siddhdnta Darpana 

diameter with surya siddhanta, though no such 
measure has been found in vedas. 

However M.B. Panta (Vedavati, Pune, 1981) 
has opined that for steller measures; maha yojana 
= 5 X Aryabhata yojana = 40 miles was used. 
Accordingly, Trisanku means 3 X 10 13 ; in 
mahayojana units it is 3X10 13 X40 miles = 207 light 
years which is really the distance of Trisanku star 
(Beta Cruris). Similarly Agastya or Argo navis has 
crossed Jaladhi or 10* 4 distance; which is 10 14 X40 
miles = 690 light years in maha yojana units (correct 
distance is 652 light years). Mandala means 
revolution or circumference, diameter is indicated 
by width or viskambha in jyotisa. Hence 72000 
yojana mandala means it is circumference. In 
mahayojana units this value means diameter of 9.1 
lakh miles which is slightly more than 8.66 lakh 
miles, the modern value. This may be correct if 
we include the corona of sun. 

Another indication of yojana measure is given 
in Rkveda (1-123-8) 

Sayana has interpreted it that dawn goes 
ahead of sun by 30 yojanas and along with it 
moves round. 

Similar verse is in RK 6-59-6 which, dawn 
goes ahead 30 steps i.e. units of length. In modern 
astronomy, dawn is taken 18° ahead, Tilaka in his 
Arctic home in vedes, page 85, has taken it 16°, 
probably for central India at 24 °N. However, in 
sandhya of each yuga, its value has been taken as 
l/12th of yuga value. Thus dawn of day time of 
12 hours is 1 hour, i.e. 1/24 of a day. This is 15° 
(360724) in angles. Thus circumference of earth is 



Lunar Eclipse 505 

30X24=720 yojanas and sun's circumfereence is 
72,000 yogana i.e. 100 times in round figures. In 
round numbers 108 japa is counted as 100 hence 
it gives almost correct dimensions of sun. 

Ratio of moon's diameter with earth's diameter 
has been slightly* increased and it is more correct 
according to modern values. Increase of parallax 
from 1/15 of earth radius to l/14th is also more 
correct and might have been confirmed by obser- 
vation. 

(2) Diameter of earth's shadow in moon's orbit 
- 85 yojanas in moon's orbit have been taken as 6 
kala i.e. lkala = 14.2 yojana. Hence linear diameter 
of earth's shadow multiplied by 6/85 gives its 
angular diameter; because it is in moon's orbit. 

Verses 16-21 - True values of diameter and 
distance — If manda kendra (of sun and moon) is 
in 6 rasis beginning with makara, manda kotiphala 
is added to trijya and substracted from trijya if 
manda kendra is in other six rasis (karka to dhanu). 
Result is substracted from double of trijya, by 
remainder we divide the square of trijya (118, 844). 
Result will be sphuta manda karna of sun and 
moon. If this method is used for star planets like 
mangala, it will give their radial distance from sun 
as centre. 

This sphuta karna in kala is multiplied by 
madhya yojana karna and divided by trijya to give 
sphuta manda karna in yojanas. Madhya bimba 
kala divided by sphuta yojana and multiplied by 
madhya yojana gives sphuta bimba kala. 

(Quoted from Siddhanta Siromani) - Manda 
karna is found like sighra karna. It is substracted 
from 2 X trijya and by remainder, we divide square 
of trijya. Result in kala is manda karna of sun and 
moon which is the distance from centre of earth. 



5Q6 Siddhdnta Darpana 

Manda karna kala multiplied by madhya yojana 
karna and divided by trijya gives sphuta yojana 
karna. Diameter of sun is 6522 yojana and of moon 
is 480 yojana (values of Bhaskara, not of this book 
- Quotation ends). 

Method of Bhaskaracarya also gives accurate 
value, still I have calculated sphuta karna from koti 
phala (instead of mandaphala because, for 3 raii 
difference between sphuta graha and mandocca, 
manda sphuta karna is equal to koti. 

Note : (1) True 
method - Madhya graha 
M is at angle from 
direction of ucca U. 
True planet S on manda 
paridhi has moved by 
same angle = ZSMN 
in opposite direction. 
SN is X on CM ex- 
tended. Figure 10 a 

NS = manda bhuja phala = r sin 6 
where MS = r = radius of mandaparidhi 
R = 3438' = OM is radius of madhya graha. 
MS' // SN is mandaphala 

MS' _ OM R , 

NS " ON " R + r cos 6 
because MN = r cos 

ON is called koti of karna, at 90° it is zero. 
Manda Karna OS = K is true distance of planet 

K 2 = ON 2 +SN 2 = (R + r cos<9) 2 + (r sin<9) 2 
= R 2 + r 2 + 2 R r cos 6 (1) 




Lunar Eclipse 507 

(2) Bhaskara approximation - His formula 

R 2 

K " 2R - K 

appears meaning less as it can be used only 
if K is already known in right side also. However 
the first K is an approximation by koti of karna 
only = R + r cos 0. This relation holds good and 
gives a better approximation from formula. 

K 2 + R 2 = r 2 + R 2 + 2 Rr cos 0+ R 2 - - - from 

(1) 

= r 2 + 2R (R+r cos0) 

= 2 RK approx neglecting r 2 

(3) Siddhanta Darpana formula has two 
unncessary steps for manda kendra 270° to 90 °, 
we first add manda koti phala to trijya, then 
substract the sum from 2 X trijya. This is equivalent 
to substracting mandakotiphala from trijya 

2R - (R + r cos 0) = R - r cos 6 

R 2 r> ,. r cos ^-i 

Now, = R (1 - — ^— ) l 

R - r cos 6 K 

r 2 cos 2 
= R + r cos 9 + + (2) 

K 

2r r 2 

Now from (1), K = R (1 ■ + — cos + ^) V2 

r 2 
= R + r cos 6 + — + (3) 

average value of cos 2 6 = 1/2, hence, expression 
(2) is almost equal to K. 

Verse 22 : Mean angular diameters (bimba 
kala of moon and sun multiplied by true daily 



508 Siddhanta Darpana 

motion and divided by mean daily motion gives 
true diameter in kala. 

Note : Linear diameter is fixed = D yojana 

Angular diameter B varies with distance, Bo 
is mean value 

Bo x R = BxK = D (1) 

True motion xK = Mean daily motion XR-(2) 
Dividing (1) by (2), we have 

Bp = B 

mean motion True motion 

„ Bo X True motion 

or B = (3) Proved 

mean motion v ' 

To prove (2), Let 0and & be the manda kendra 
for today and tomorrow at sunrise 

True longitude for sunrise today 

Rsinfl x R 



(Fig. 10a) 



R sin ' x R 



= Apogee today + arc , , 

I manda karna today 

True longitude tomorrow sun rise 

( R sin ' : 

= Apogee tomorrow + arc 

manda karna tomorrow 

Taking difference of these two equations 

Daily motion for today = Daily motion of 
apogee 

(0 - 0) X R 
manda kendra for today a PP rox \ ) 

Here, manda kendra difference in one day 
has been ignored, (0'- 6) = daily motion of manda 
kendra i.e. mean daily motion. 

This is formula (2), if we ignore very slow 
motion of apogee. 



Lunar Eclipse 509 

Verse 23 - Formula are 

Diameter i n yojana x R 

fa) Diameter in kala = — ~-j : 

w Spasta karna yojana 

This follows from (1) above. 

(b) B = Bo ± gatL £ Q f o r sun 

Addition is done when manda kendra is in 
2nd and 3rd quadrant, otherwise substraction is 
made. 

_ B True motion 

Proof : 



or 



B Mean motion 
B - B True motion - mean motion 
B mean motion 

gati phala 



mean motion 



Bo x eati phala 

or B - Bo = 7 

mean motion 

_ gati phala x 32/32/8 

mean motion (59/8) 
(Putting values of Bo and mean motion) 
gati phala x 11 
20 
approx if both are in kala 
If gatiphala is in vikala then, the correction 

gati phala x 11 _ gati phala 

20 x 60 " 110 a PP rox * 

Verse 24 : For moon 

moon eati phala , . 
b = bo ± ^^ (c) 



is 



510 Siddhanta Darpana 

Proof - As in above formula correction is 

b x gati phala gatiphala x 31/20 

mean motion 790/35 

= gatiphala/25 approx 

,^ „ True sun speed x 11 
(QB = J? 

^ v , True moon speed - 7 
Proof (i) Formula (c) is obvious 

Bo x True speed 32/32 

B = * = True speed x 

Mean speed r 59/8 

= True speed X £ approx. 

_ bo x True speed _ True speed x 31/20 
W ~ Mean speed ~ 790/35 

True speed x 31/20 True speed - 7 
= 31/20 x 25 -f 7 = 2^ a PP rOX 

as 7 is very small compared to speed of about 
800 kala per day. 

Verse 25 : Shadow length of earth (conical 
from centre) 

True sun karna x diameter of earth 
Sun diameter - Earth diameter 




v 



Figure - 11 



Lunar Eclipse 



511 



Note : In figure 11, BE is Parallel to AS, so 



AS SV SE + EV 
BE : 



SE 



+ 1 or EV = 



SE x BE 



EV EV EV AS - BE 

Verse 26 : According to Siddhanta Siromani 

Diameter of earth's shadow in moon's orbit 

= Earth diameter — 

(Sun diameter -Earth diam) X moon distance 

sun distance 

Note : This is called reduction in earth's 
diameter; because sun is bigger and earth's shadow 
converges into a cone. 




Figure - 12 



In fig 12, S,E, M are centres of sun, earth 
and shadow, Common tangent line ABC meets 
SEM at V. Radius of sun, earth, shadow are R, a, 
e. Distance of sun and moon from earth are r, n 
Shadow cone from moon MV = x. 

Now in similar triangles ASV and BEV 

AS BE 



SV 



EV 



or 



AS 
BE 



SE + EM + MV 
EM + MV 



R r + ri 4- x 

or — = 

a n + x 



n + x 



+ 1 



512 Siddhanta Darpana 

ri + x a ar 

or = or x = n - (1) 

r R-A R-a w 

In similar triangles BEV and CMV 

BE EV EM + MV 

CM " MV MV 

a ri + x r a 

or - = = — + 1 

e x x 

ri a " e eri ,~ 

or — = or x = - - -(2) 

x e a - e 

Equating values of x from (1) and (2) 
ar e ri 



- ri = 



R-a a - e 

e _ ar _ ar - n (R - a) 

° r a- e = n (R - a) 1 n (R - a) 

a ^ n (R - a) ar 

or — = 1 + 



e ar - n (R - a) ar__ n (R — a) 

ar-n(R- a) R-a 

or e = = a x n — (3) 

r r 

After multiplying by 2, result is proved. 

Verse -27 Earth shadow in kala 

Moon true motion 

7 
Sun true motion x 78 

145 

Note : Formula (3) in previous verse can be 
written as 

2e 2a 2R - 2a 



ri 



(1) 



Lunar Eclipse 513 

This gives angular diameeter in radians. 
Multiplied by Trijya = 3438' it will give diameters 
in kala. First term in kala in right side of (1) is 

2a x 3438 

ri 



ro True speed 

But — = * 

ri mean speed of moon 

(See equation (4) after verse 22), r = mean 
distance 

Hence this becomes 

2a x 3438 X True speed 

r x mean speed 

1600 x 3438 

= " 48,705 X 790/35 X 1me Speed 
(Because 2a = 1600 yojana 

ri = 48,705 yojana, 

mean speed of moon = 790/35 kala 

True speed 
= -j — — approx. 

Second term in (1) is similarly 

(2R - 2a) x 3438 x True speed of sun 

Mean distance x mean speed of sun 
= True speed 
v (72,000 - 1 600) x 3438 

7608, 294 X 59/8 &** ValueS 

78 
= 145 a PP rox * 

Hence the formula 



514 Siddhanta Darpana 

Veieses 28-30 : Meaning of rahu 

Lunar eclipse is caused when moon enters 
earth's shadow, and solar eclipse is caused by 
covering of sun by moon. This is possible only 
when sun, moon and shadow of earth are near 
node (pata) of candra whose names have been 
given rahu and ketu (half part of rahu itself). Hence 
it is said that rahu devours sun or moon in eclipse. 

In siddhanta siromani - If eclipse is caused 
by same rahu, why there are different direction 
(of beginning of) eclipse, different times (short or 
long periods) and different coverings (total or 
partial eclipse). So persons assuming eclipse by 
rahu have false pride of their knowledge of sphere/ 
actually they are fools and against (true meaning 
of) Veda, purana and samhita. 

Rahu is shadow planet (a fictitous point), 
which covers moon by entering earth shadow 
(being near it); and covers sun by entering moon. 
For such ability, sun has given boon to him. This 
type of interpratation is not against scriptures. 

Veerse 31-32 : Reasons of eclipse 

Lunar eclipse - Shadow of earth is in opposite 
direction of sun and moves east wards in ecliptic 
like sun. At the end of full moon, when moon is 
in opposite direction of sun, its speed is more than 
shadow speed, so it enters the shadow and crosses 
it. After entring shadow, its light (from sun) is 
lost. Thus lunar eclipse is seen. 

Solar eclipse : At the end of amavasya, when 
moon and sun are in same direction (same rasi), 
then moon moving east covers sun and with faster 
speed crosses out in east direction. Sun being very 



Lunar Eclipse 515 

luminous cannot bee seen. While covering, it, only 
lightless moon is seen. 

Verse 33 : Digamsa correction of rahu 

As we add or substract fourth phala (slghra 
correction) in pata of star planets like mangala, 
similarly digamsa phala of rahu is calculated as of 
moon and it is added or substracted. 

Note : Digamsa phala = 1/10 of mandaphala 
of sun. This is correction in moon's orbit due to 
variation in annual attraction of sun, which changes 
the direction of moon's orbit. Hence it changes 
direction of rahu also. This correction has been 
described in chapter 6. 

Verses 34-35 - Sara of moon 

Sphuta pala is deducted from sphuta moon' 
at corrected parvanta time. Bhuja jya of this arc is 
calculated. This is jya of viksepa kendra. We add 
1/38 part of it. From half of result, arc is found. 
This arc divided by 6 gives sara of moon. 

When moon - pata (or vipata candra) is in 
six rasis beginning with mesa, sara is in north 
direction, otherwise in south direction. 

N N' m S 




Q Q' 



Figure - 13 

Notes - Maximum sara of moon has been 
stated as 309' in siddhanta darpana i.e. inclination 
angle e = ^MNS is given by R Sin e = 309' NS 



526 Siddhanta Darpana 

is ecliptic on which S is position of Sun or earth's 
shadow. NS = m = distance of moon from node 
N. NM is orbit of moon with moon at M. Its sara 
is MS = p. 

p = m tan e as AMSN is right angled and 
almost plane due to small size. 

, m . R sin e m . R sin 309 

US P R cos e R sin (5400 - 309) 

m x 308 m ' „ x 

= ~^iii — = — r approx - ' ■ (1) 

11 + - 

3 

However, at the time of eclipse, S has slow 
motion and is considered fixed and we calculate 
only the moon's speed. Relative speed of moon is 
obtained by adding vector VV 1 equal and opposite 
to motion of S to velocity vector MV of moon. 
Thus resultant motion of moon is smaller and in 
direction MV'N' which make angle e' slightly bigger 
thane' 

VQ motion of sara 



tan e = 



tan e' = 



QM motion along ecliptic 
VQ 



or 



QM - VV 
tane' QM 



tane QM - W 

790/38 X cos (309) 



or tan e' = tan (309') „ nn ,^ n ,„ rtrtX 

v ' 790/38 cos (309) - 59/8 

thus e' = 333 = 5° 33' 
Shortest distance of moon from ecliptic 
= p cos e' 
= SP which is pependicular from S to MV' 



Lunar Eclipse si7 

Hence effective sara = p cos e' 

_ p . R . sin (5400 ^ 333) P x 3420.5 

R " 3438 (2) 

Equation (1) takes value (3438/3423) times 
more than the sine value. For half the angle 
increase is about 1/38 times as approximated here. 
Hence after increase of 1/38 in m, Sin,, of its half 
value in taken, then divided by six again. Taking 
sine almost equal to small angles, formula given 
is 

39 1 13 

P = ^8 X U m = m m 

which is almost same as (2). as may be 
verified. Due to relation (2), the effective inclination 
of moon's orbit is reduced by about 18' to 290' 
approx. Hence the value of parama sara was taken 
as less than the true value, in earlier texts. 

Verse 36: Method for sara gati 

Instead of finding arc, we multiply the 
kotiphala of moon and it is multiplied by pata and 
motion of moon and divided by trijya. Result is 
current speed of sara. By adding or substracting 
kranti gati, we get sphuta sara from equator. 

Notes: (1) There are three confusing words in 
the verse-Whose kotiphala is to be taken is not 
specified — I have interprated it to be kotiphala of 
moon's movement along ecliptic i.e. its rasi etc 
from pata. 

Whether motion of pata and moon both are 
meant — 

pata has very small motion and when motion 
sun is being neglected, much smaller motion of 



518 Siddhanta Darpana 

pata cannot be taken. Hence it is pata or sara from 
ecliptic and motion of moon. 

Result of this multiplication and correction 
with kranti both are called sphuta sara. First sphuta 
sara is distance of moon from kranti vrtta. Second 
sphuta sara is distance from equator = distance 
frojm kranti vrtta(sara)+ distance of spasta moon 
on kranti vrtta from equator (i.e. kranti of moon). 

Translation has been made according to these 
clarifications. 

(2) Sphuta sara from equator has already been 
explained. Now p=sin m. tan e. 

p=sara, m= distance of moon from pata along 
ecliptic £ = angle of inclination of moon's orbit with 
ecliptic, since e is constant, taking differentials 

Cos p. (3p= Cosm. <5m. tan e 
Here dp and (5m are motion of pata and moon 
in unit time of hour or day. We are to find dp. 

Rcosm K tan £ 

dp = ~ .(5m. — — (1) 

r R cos p 

Now cos p= cos e/sin m according to Napier 
relations of right angled triangle N E P. 

tan £ 

Hence, = sm m sin e ~ sin p ~ p 

cos p 

Hence (1) becomes 

motion of pata 

Kotiphala of moon x Moon gari ,x Pata 

Tnjya 

verses 37-38: sara from chart 

Pata is substracted from moon and bhuja of 
the resulting angle is found. From degrees of bhuja 



Lunar Eclipse 519 

sara etc can be found in charts where sara, sara 
difference, kotiphala and bhujaphala etc have been 
given in appendix: These have been given at 
intervals of 225. 

Alternatively, pata is substracted from moon. 
Its bhuja is converted to kala. Its 1/16 will be 
subtracted and divided by 11. (i.e. moon-pata). For 
greater values, this will be incorrect 

Notes : Kala of bhuja is almost equal to jya 
for small angles. 

It is to be multiplied by 

309 Parama Kranti 



3438 1,c " 
1 


Trijya 
1 


-., 39 
309 


1 
11 + 8 



approx. 



Substraction of 1/16 part is to convert the 
bhuja approximately to its jya: 

Verse 39: Extent of eclipse (grasa) 

The planet which is to be eclipsed is called 
grahya or chadya and the planet or shadow which 
covers it, is called grahaka or chadaka. Half the 
sum of their angular diameters is 'manardha' or 
'manaikyardha' . If sara is more, there cannot be 
eclipse. Differance of manardha and sara is the 
grasa (covering). If grasa is more than grahya, then 
eclipse is total. Remaining part of grasa is in sky. 

Note : Except for the terms, the cause of 
eclipse has already been explained. 



520 



Siddhdnta Darpana 




M 




Figure 14 (a) Figure 14 (b) 

P is centre of planet to be eclipsed and XM 
is diameter. Its distance from centre of coverer O 
is sara-OP. Either the coverer (chadaka) earth 
shadow or covered (chadya) sun is on ecliptic 

In figure 14(a) covered portion 

XY = PX - PY 

= ri - (OP - r 2 ) = ri + r 2 - OP 

Where rl and r2 are radii of the bimba. 

When XY > 2ri then eclipse is total as in fig 
14(b) 

Verse 40 : Direction and stages- 
Lunar eclipse has contact in east and end in 
west often (as explained in verses 31-32) Calculation 
is done as per this rule only. But solar eclipse 
often starts in west and ends in east direction, 
However, sometimes south west direction is calcu- 
lated instead of east west direction 

When grahaka just completely covers the 
grahya, it is called 'nimilana' time. When it is about 
to start emerging, it is called unmilana. Time from 
nimilana to unmilana is called 'marda kala' or 
'vimarda kala' (period of total or maximum eclipse). 
Total time of eclipse from sparsa to moksa is called 
grahana kala or 'sthityardha' kala. 



lunar Eclipse 521 

verses 41-43 : Total time and time of complete 
eclipse- 

Parvanta time. When moon and sun have 
same rasi etc. (in solar eclipse) or their difference 
is exactly six rasis; it is the time of eclipse. This 
time (after minor correction of verse 11) is called 
sama parva kala and is middle point of eclipse. 

Square of manardha (half sum of diameters 
of grahya and grahaka) is substracted from square 
of sphuta sara and of remainder, square root is 
taken. This is multiplied by 60 and divided by 
difference of moon gati and sun gati. Result is half 
time of eclipse (sthityardha kala). Its double is total 
time of eclipse. 

Similarly, square of half the difference of 
diameters of grahya and grahaka is substracted 
from square of sphuta sara. Square root of 
remainder is multiplied by 60 and divided by 
differance of gati of moon and sun. Result is 
'marda- ardha' kala in ghati etc. Its double is 
'marda' kala or time of complete eclipse. 

From samaparva kala (mid point of eclipse), 
substraction of sthiti ardha and marda-ardha give 
sparsa and nimilana times. When sthiti ardha and 
marda-ardha are added, it gives unmllana and 
moksa times: 

For more correct times, we calculate sphuta 
candra and sara at time of sparsa, unmllan etc and 
from them again these times are calculated. In solar 
eclipse, repeated larhbana corrections are made. 

Notes : (1) This is an approximate formula in 
which sara of moon is considered to be same, 
hence there is need for successive approximation. 
First, we derive the approximate formula, assuming 



522 Siddhdnta Darpana 

the sara to be minimum distance of moon from 
mid point of eclipse. 




Figure 15 - Period of eclipse 

In figure 15, O is centre of earth shadow and 
ON is ecliptic. Shadow is considered fixed and 
moon is moving in direction Mi N with relative 
speed (moon-sun). This direction is slightly more 
inclined 5° 33' compared to 5° 9' angle of moon's 
orbit with ecliptic, as explained in verse 35. Ml, 
M2, M3, M4 are positions of moon at 1st contact 
(sparsa), 2nd (nimllana), 3rd (unmilana) and 4th 
contact (moksa) 

OP is perpendicular on Mi N and is almost 
equal to sara. This value of sara is assumed for all 
the four positon. Right angled triangle OPMi is 
almost a plane figure. 

Hence 

M i p = VMi O 2 - OP 2 

= V l(moon bimba + shadow bimba) z - sara 2 
2 

Moon moves along MiP with relative speed 
of m-s where m and s are daily motions of moon 
and sun(or shadow) in 60 danda. 

u •. mi „„. 60xMiP 

Hence it will cover MiP in danda 

m — s 

This is same as M4P distance (as Mi 0=Mi 
O). 



523 

lunar Eclipse 

Similarly, OM 2 or OM 3 = radius of shadow- 
radius of moon. 

P M 2 or PM 3 = VoNg - OP 2 



I "— m 

= V 1 (shadow diam - moon diam.)* - sara' 

2 
Hence half time of total eclipse 

60 

m — s 



— -~ ^manantara 2 - sara 2 

— C 



(2) Let T be time of conjunction, when moon 
and earth shadow have same longitude, and p the 
latitude (sara) of moon, North latitude is considered 
positive, p' is hourly increase in latitude (increase 
towards north is positive) 

m'= excess of hourly increase in longitude of 
moon over that of sun. 

M= angular radius of Moon, S= angular radius 
of shadow at the moon. 

Then at any time t hours after time of 
conjunction, T, the distance between shadow and 
moon in longitude is m't and the latitude of moon 
is p+p't. 

Thus distance between centres of shadow and 

moon 

={m ,2 t* + (p + P' t) 2 l 1/2 

The eclipse begins or ends when the moon's 
rim just touches the rim of the shadow in entering 
it or leaving it. Distance between such time is 
S+M=D say (fig 15) then { m' 2 t 2 + (p + p' t) 2 } m 
= D gives the time of beginning of eclipse. Solving 
this for t, we get 



524 



Siddhanta Darpana 



t = 



pp 



m /2 + p' 2 



« 2 w 2 
P P 



./2 



,ia 



+ 



(m" + p' z ) 



D 2 -p 2 
m' 2 + p' 2 



iv 2 



The + sign gives the end and -sign (earlier 
time) gives the beginning. 

Total phase of the eclipse begins or ends when 
the rims touch the moon being inside shadow (M 2 
M3 position of fig. 15) i.e. D= S- M, Putting this 
value of D in above soulution, we get the times 
of beginning or end of total phase of eclipse. 

Discussion of results: 

(1) The eclipse begins at 



T- 



PP 



m' 2 + p /2 



P 2 P' 2 



,2 



ao. 



+ 



(m" + p") 



D 2 -p 2 

m' 2 + p' 2 



Vi 



(2) Eclipse ends at 
PP 



T- 



m' 2 + p' 2 + 



p 2 p' 2 



rr + 



D^-p' 



Vi 



(m' 2 + p' 2 ) 2 m' 2 + p' 2 j 



hours 



For full eclipse time D=S+M. For total eclipse 
D=S-M. r 

(3) Middle of the eclipse falls at 
PP' 



T 



'2 



7^ hours 



m' z + p' 2 

(a) If p and p' are both positive or both 
negative, middle of the eclipse is before the time 
of conjunction. 

(If one is positive and other negative, middle 
is after conjunction) 

(b) Only when the latitude at conjunction p=o, 
the middle falls at T, the time of conjunction, 
because p' cannot be zero near a node. 



Lunar Eclipse 525 

In verse 11— p latitude is positive then 
udayantara is positive, p' i.e. speed of latitude is 
positive when bhujantara is positive (verse 36). 
Thus when both are positive or negative time is 
corrected 

(4) If D=p, (1) and (2) reduce to 

t PP' -+. PP' 

1 — 1 n — 



m' + p' m' + p' 

(a) Eclipse begins or ends at the conjunction 

(b) Duration of eclipse is — 2 2 which may 

m' + p' 

amount to about 22 minutes 

(5) The duration is zero, when the expression 

between the double brackets is zero, i.e. p is greater 

D' 2 

than D by , 2 &r , (neglecting fourth powers 

of p'/m'), which may amount to about 14" in the 
mean. 

(6) If t is not real, there is no eclipse, or total 
eclipse, according as D is taken to be S+M or S-M. 
Then p-D is more than about 14" calculated above 

(3) Conditions of eclipse in equator coordinates 

Instant at middle of eclipse is chosen as origin 
of time in figure 16. 




Equator 

Figure 16 



526 Siddhanta Darpana 

Equatorial coordinates of centre C of shadow 
at time t hours is a, d and of centre M of moon 
be ai, di. Then, if P is the pole and M D the 
perpendicular from M on P C (on celestial sphere), 
CD= <5i-<3 and DM= (ai-a) cos <5i, nearly. So 
CM 2 = (di - a) 2 + (ai - a) 2 cos 2 <5i - - (1) 

If, hourly rates of increase of a, ai, 6, d\ at 
t =o are (a) o, (ai) (6) (^i)o respectively, we can 
write (1) as 

CM 2 - [{ (dOo + d\ t } - [(6)0 + <5' t}] 2 

+ [ {(al) + a' t } - { (a) + (a' t) } ] 2 cos 2 <3, (2) 

approximately, neglecting the changes in cos 
(3i due to changes in 6\, because cos 2 6\ in the 
above equation is multiplied by a factor which is 
small. Equation (2) is of the form 

CM 2 =a t*+ bt +c (3) 

Where a, b, c are known quantities. 

51 

If we put CM= — (P + Pi - S) + S 2 the two 

values ti and ti given by (3) are the times of 1st 
and fourth contacts (sparsa and moksa). For the 
second and third contacts (i.e. beginning and end 
of totality, we put 

51 

CM = — (P + Pi - S) - Si and solve for t. 

w/V/ 

1 



Middle of the eclipse is — (ti + ti) 



2 K± *-' 2a 

verses 44-45 —Single time calculation 

Method above uses successive approximation. 
Now method of single time calculation is described. 
Sara of samaparva kala is calculated. Its half is 



Lunar Eclipse 



527 



divided by difference of gati of moon and sun; 
Result in danda etc is substracted from samaparva 
kala sara, if sara is increasing, otherwise it is added. 
For this parvakala, new values of moon and its 
sara are found. Diffenence of this sara and 
samaparva kala sara in vilipta is squared and its 
half is taken. Its square root substracted from 
samaparva kala sara is sphuta sara. Sthiti kala 
calculated from this is correct. Now more accurate 
value of shadow is stated (in verses 78-84) 

Notes — In figure 16(a). 

AB is ecliptic and CD is moon's orbit, relative 
to shadow centred at S on the ecliptic. S and M 




B 
A N2 N1 S 

Figure 16a - One time calculation of sthiti ardha 

are the centres of shadow and moon respectively 
at the time of oppositon. SMi is perpendicular from 
S on moon's orbit and MiNi is perp. from Mi on 
the ecliptic. Then Mi is the moon's centre at the 
middle of the eclipse. AMMiS is almost plane, 
^MMiS=90 

MS = moon's latitude at opposition, 

Z.MSM1 = i, inclination of moon's orbit to 
ecliptic. 

Ni S = Mi M approximately as i is small 

= 3 ° 9 * MS minutes (Kala) as R sin i = 309' 
3438 v ' 



528 Siddhanta Darpana 

309 x 60 x MS 1 , j 

_ TTTr x : — ; : danda 

3438 gati antara (of moon and sun) 

MS . 

— : approximately (1) 

2 x gati antara vr J 

_ MS x 309 x 60 x 60 
or Ni S - 343g (790 , 35 „ _ 59 , 8 ,^ pa a 

MS i n\ 

= — palas (2) 

This time is subtracted from the sthitiardha. 

Since square of sara is used in calculation, 
average of squares of sara at M and Mi is taken. 
Hence, half the square of difference is taken. 

verses 46-50 : Grasa from time. 

Now method is described to calculate grasa 
from time and vice versa. If time is before mid 
eclipse, it is substracted from sparsa sthiti ardha 
time. Remainder in danda etc. is multiplied by hara 
=(moon gati-sun gati) corrected for lambana for 
solar eclipse, next chapter verse 46-47) and divided 
by 60. This will be koti kala of lunar eclipse. 

In solar eclipse, it is again multiplied by 
madhya sthiti ardha and divided by sphuta sthiti 
ardha, to get sphuta koti kala. 

For given time, squares of koti kala and bhuja 
kala are added, square root of sum is karna. This 
karna substracted from half the sum of bimba kalas 
gives grasa. 

If the given time is after mid eclipse time, it 
is deducted from moksa sthiti ardha. Difference is 
multiplied by gati antara of sun, and moon (hara) 
and divided by 60. We get koti. Then sara of given 



lunar Eclipse 529 

time is found, from which spasta koti kala of solar 
eclipse can be found. Again karna is found by 
adding the squares of bhuja and kotikala and taking 
square root. Karna substr acted from half sum of 
bimba, gives grasa. 

From grasa value, remaining free portion of 
eclipsed planet can be found. 

Notes . (1) Grasa = covered part (literal, 
meaning devoured portion) 

Amount of grasa is the length of diameter 
along the line joining centres of covered and 
covering discs, which has been eclipsed. 

Magnitude of eclipse (in modern astronomy) 
is grasa expressed as fraction of diameter. Thus 
grasa = radius of shadow + radius of moon 

-distance between centres of shadow and 
moon 

Magnitude = grasa/diameter of moon. 

For solar eclipse, instead of shadow, we take 
moon's disc and covered disc is of sun. 

(2) Formula of grasa has already been estab- 
lished while calculating sthiti or marda times. To 
revise, refer to figure 15. If M is any position of 
moon's centre, MP is distance covered from central 
point P. If it is before P, then it is at Mi (contact 
point or sparsa). Then in the given time after 
sparsa, moon moves from Mi to M. The remaining 
portion is MP till mid time at P. 

Thus in time (sthiti ardha-given time) =t 

r 

planet covers MP which is — — — where m' is 

60 

difference of daily speeds of moon and sun. 



530 Siddhanta Darpana 

m/60 is speed in one danda. MP is koti kala 

OM = Vqp 2 + MP 2 = V m't 2 ^ = Kama 

U1 ^ 1VU (~7Z) + (sara) 

6U 

1 

When OM < — difference of diameters, com- 
plete portion of moon is covered. For OM bigger 
than this value l/2(sum of diameters)— OM is 
amount of grasa. Similar calculation is done for 
period after midtime. 

(3) In solar eclipse there is fast change in sara 
and valana, hence true kotikala is found. 

spasta koti kala _ madhya sthiti ardha 

Madhya koti kala spasta sthiti ardha 

Because, if sthiti ardha increases, the dif- 
ference with given time decreases and koti kala 
decreases. Thus, they are inversely proportional. 

Verses 51-53 : Time from grasa 

When grasa is between sparsa and mid time, 
then it is substracted from half sum of covered 
and covering discs. This gives difference karna 
between centres of two discs. From square of this 
karna, we substract square of spasta sara at that 
time. Square root of difference will be koti kala. 

For solar eclipse this kotikala is multiplied by 
lambana corrected sthiti ardha and divided by 
madhya sthiti ardha. This gives spasta koti kala. 
This is multiplied by 60 and divided by difference 
of daily speeds of moon and sun. Result in danda 
etc. is the time after sparsa. 



Lunar Eclipse 531 

9 

For position in second half of eclipse, the 
result is substracted from sthiti ardha time, The 
remainder will be time remaining till moksa. 

Notes : This is reverse process of the previous 
method and uses the same formula. 

Verses 54-Method for solar eclipse-For solar 
eclipse, the sthiti ardha for sparsa and moksa is 
called mean sthiti ardha, because special paral- 
lax(lambana) correction is done in this. Hence, all 
processes are done with mean sara (this doesn't 
change in short period of eclipse). Repeated 
parallax correction will give correct time. 

Verses 55-59 :Direction of eclipse from parallax— 

Now I describe valana(parallax) correction in 
kalas for correction of moon and sun in their 
eclipses, which arise due to ayana and aksamsa. 
Due to these effects, direction of sparsa, mid-point 
and moksa of an eclipse is known in east or west 
portions (kapala) of sky. 

In case of lunar eclipse, sayana candra, and 
in case of solar eclipse, sayana sun is found. Its 
kotijya (in kala) is multiplied by parama kranti 
(1410) and divided by 3 rasis (5400kala). Result is 
ayana valana. This is in same direction (east or 
west part of sky) in which eclipse takes place. 

In solar eclipse, rasi of sun and moon is same, 
hence ayana valana can be found only from moon. 

We calculate the nata kala in pala from moon 
midday in lunar eclipse and from solar midday in 
solar eclipse This multiplied by 90° and divided 



532 Siddhdnta Darpana 

by its half day time gives nata in east or west 
direction in degrees 

This nata is multiplied by aksamsa of the place 
and divided by 90. Result will be aksa valana in 
north direction for east nata direction or valana in 
south direction for west nata. 

Aksa and ayana valana are added when in 
same direction and difference is taken for different 
directions. Result will be dik- valana in degrees of 
moon in lunar eclipse and of sun in solar eclipse. 
This is true valana from which sparsa and moksa 
directions can be known. Its measure in angula 
has been stated while describing parilekha 
(degrees). 

Notes (1) Sphuta valana is the angle between 
east or west point of disc of eclipsed planet with 
kranti vrtta (ecliptic). This is made of two com- 
ponents. Due to aksamsa of the place (distance 
from equator), kranti vrtta cuts the horizon in 
eastern half of sky in north direction from east 
point. So ecliptic is towards north of east point of 
disc in east half of sky (and towards south in west 
half). This is called aksa valana. 

Due to angle between ecliptic and equator 
(causing ayana), ecliptic is inclined further towards 
north when say ana makara is on meridian (north 
south vertical circle). When sayana makara is on 
meridian (±90°), it is shifted southwards in east 
half of sky. For west half, the directions are 
opposite. This component is called ayana valana. 



Lunar Eclipse 



533 




E' E K 

Figure 17 - Ayana and Aksa valana 

NVZS is yamyottara vrtta (meridian) at desired 
place. 

NES is east half of horizon (ksitija vrtta), 
north, east and south points shown 

Z=kha-swastika (Zenith), 

ZE = Samamandala, Z'E' parallel to ZE 
through C,- 

e, w — are east and west points. 

KK' = Kranti vrtta 

C= centre of planet disc to be eclipsed 
(chadya) 

NCS = Samaprota vrtta of c (circle of position) 

V = North Pole in sky 

P = pole of ecliptic (kadamba) 

P P1P2 is kadamba vrtta in which P moves 
round V in a day. 

Z C 1 =Nata degree (in time units) of C 

P2 = Kadamba when sayana karka is at K' 
(meridian) 

Pi = Kadamba when sayana makara is at K' 
CV = Polar distance, C P = Kadamba distance 



534 Siddhanta Darpana 

ZNCV = = Aksa valana, ZVCP = 0'= Ayana 
valana = Ayana valana 

L NCP = 6 + 6' = sphuta valana = L KCE' 

In this figure 17, for position P betweeen 
± 90 Q distance of Pi, ayana valana 6' is also in 
north direction, hence sphuta valana is 6 + & as 
shown in figure. For P between ±90° of P2, 6' will 
be in south direction and sphuta valana will be 
6-6'. The direction of valana will be opposite, when 
planet is in west kapala (west half of sky). 

(a) Aksa valana - From spherical triangle NCV 

Sin NCV _ sin CNV sin ZC 

sin NV sin CV sin (polar distance) 

because NZ and NC both are right angles, 
hence angle between them is equal to ZC, which 
is natamsa of planet. 

sin (polar distance) = R cos 6, (d - kranti of 
planet) 

= Dyujya or radius of ahoratra vrtta. 

sin NV = R sin O , O = aksarhsa of the planet. 

so, 

R sin x sin ZC R sin x sin ZC 

sin NCV = ^— r-- = £ j (1) 

Dyujya R cos o 

Rule for finding natamsa— 

This is as per Bhaskaracarya. In half day or 
half night time, a planet rises 90° from horizon, 
hence nata kala multiplied by 90 and divided by 
half day (or night in lunar eclipse) time gives 
natamsa in degrees. This is not the angle from 
vertical Z point, but the angle between meridian 



lunar Eclipse 535 

and samaprota vrtta, corresponding to nata kala. 

(H) 

Its relation with natamsa from Z is z = ZC. 

This can be found from spherical triangle NZC, 

cot Z C x sin ZN 

=cos ZN. Cos NZC + Cot ZNC. sin NZC 
But ZN = 90° hence sin ZN =1, cos ZN = 
Hence, cot ZC = cot ZNC. sin NZC 

^^ cot ZC 

or, cot ZNC = — — rr=^ 
' sin NZC 

or, tan ZNC = Sin NZC. tan ZC 

But, L NZC = 90° + L EZC = 90° + agra 

Hence sin NZC = cos (agra) 

Hence tan ZNC = cos (agra) x tan z 

Rule for aksa valana : 

Surya siddhanta and Bhaskara II both have 
given the formula (1) i.e. Jya of natakala is 
multiplied by Jya of aksamsa and divided by dyujya 
or semi diameter of diurnal circle. 

In this text, R sin NCV and R sin <p both 
have been approximated to the angles NCV and 
<p and dyujya is equated to 90°. When d is small, 
R cos 6 = R sin 90°. nearly. Thus all the 4 jya are 
slightly increased to the arcs and the errors almost 
cancel each other as a rough rule. 

(b) Ayana valana is known from spherical 
triangle PCV in figure 18 

Sin L PCV ■_ .sin CPV 
sin PV " sin CV 

* 

sin PV x sin CPV 

or sin PCV = : — ^77 

sin CV 



536 



Siddhanta Darpana 




Figure 18 



PV= distance from dhruva to kadamba which 
is equal to parama kranti (angle between ecliptic 
and equator). CV is distance from dhruva whose 
jya is kotijya of kranti 

LQ P V is the 
angle between circles 
from C to ecliptic pole N2 
and ay ana circle Ki P. 

Positions of 

planet on ecliptic and 
equator are Li and L 2 . 

L CPV=arc Ki Li 
= 90° - ML! where M 
is vernal equinox. 
Hence Jya of CPV is kotijya of MLi = sayana graha 

Jya of parama kranti x kotijya 
Kotijya of kranti 

This is the formula given 

Verse 60-65 : Period of lunar daytime: 

Solar day time has already been described. 
The period from moon rise to moon set is its day. 
At sunset time, sphuta sayana sun and moon are 
calculated. For sun, rising time (in asu) is calculated 
for remaining part of rasi and for moon, it is for 
lapsed part of rasi. These two udaya asu are added 
with rising times (udaya asu) of the rasis from sun 
to moon. We add 56 asu to the total and divide 
by 360 to make them ghati. This time after sun 
rise moon will rise. 

For finding moon-set time, sayana sun and 
moon for next sun rise time is calculated. Rising 
time of rasis between sun and (6rasi + moon) is 



Thus sin PCV 



lunar Eclipse 537 

calculated and 56 asu lambana time is substracted. 
This time after sun rise, moon will set. 

Sara of moon is very little (within 5° 9' and 
almost zero at eclipse time). Hence time between 
rising and setting of sun will be its day time, which 
has been calculated for diurnal circle of sun. 

Sara in kala (minutes of angle) at rising time 
or setting time is multiplied by palabha and divided 
by 12. Result will be added to rising time if sara 
is south and substracted if sara is north. Reverse 
is done for correcting moon-set time. Thus we get 
sphuta time of moon rise and moon set. 

Alternatively, sphuta gati of moon at midnight 
is divided by 19 and result in pala is added to 
night time of sun. This gives day time of moon. 

Notes : (1) Difference between rising times of 
sun and moon is the difference between rising 
times of their rasis, since sun and moon move in 
almost same ahoratra vrtta. Sara at eclipse time is 
almost zero. 

Since, at purnima time, moon-sun is less than 
180° (it is 180° at end of purnima), when sun has 
risen, moon will be slightly above west horizon. 
Thus difference of moon from sun +180° or (moon 
+180°-sun)distance is to be covered by moon for 
setting after sunrise. 

Due to parallax, angle of moon at horizon 
seen from surface is 56' lower than the angle 
calculated from earth's centre. Thus moon will rise 
on horizon after covering 56' more. Hence moon 
rise time will be later than the rising time of 
rnoon-sun by further rising time of 56'. 



538 Siddhanta Darpana 

Similarly setting of moon will be earlier by 
corresponding rising time of 56' extra arc. 

(2) Alternative formula — Solar day in asu is 
more than naksatra day in asu (21, 600) by the 
daily motion of sun (59.8"), extra time taken by 
earth to cover this distance covered by sun in mean 
time. 

Similarly, lunar day is more than naksatra day 
by its daily motion in asu i.e. 790735" It is more 
than solar day by 790 (1-1/13.37) asu For true speed, 
it is more than solar day by moon gati (1-1/13.4) 
asu, relative speeds of sun and moon assumed 
almost content 

Due to parallax the decrease in day time is 
(moon gati/14) both at moon rise time and moon 
set time. Hence (moon day-sun night) 

moon gati ' 1 ^ ^ - 



l - 
l 

2 

/ 



13.4 



2 x moon gati 



asu 



(Moon day = - moon - day and night ) 



= moon gati 



12.4 



\ 



= moon gati x 



2 x 
6 



13.4 
6.2 

13.4 



1 

7 



\ 



asu 



7 



V 



pala 



moon gati 
= V — Pala approx. 



Verses 66-69 : Explaination of valana correction. 

On great circle from north pole to south pole 
in the sky, pole of ecliptic called 'Kadamba' is 
situated 23° 30' south from north pole. This is 
surface centre of ecliptic in north part of celestial 
sphere. 



Lunar Eclipse 539 

The south surface centre of ecliptic (kranti 
vrtta) is called 'kalamba' which is north from south 
pole by same 23° 30' angle on ayana prota vrtta 
(between two 'dhruva') 'Sara' is calculated along 
kadamba prota vrtta which is distance from ecliptic. 

Moon disc moves fastest of all the planets. 
Hence only its difference along two circles ayana 
prota and kadamba prota is calculated. 

Distance of moon from 'dhruva' on dhruva 
prota vrtta and from 'kadamba' along great circle 
through kadamba is taken. Their difference (an- 
gular) is multiplied by 360 and divided by 
circumference of moon disc (angular). This gives 
ayana valana. When moon is in north ayana, it is 
north valana and it is south valana in ' south 
hemisphere from equator. 

For aksa valana, Lalla and Srlpati have 
calculated versine of nata. But it has been done 
from R sine of nata by Brahmagupta and Bhaskara 
II. For ayana valana also two methods exist. One 
is from kotijya of madhya graha and the other 
from versine of bhuja of sayana graha. But in my 
method, no jya is needed because nati is according 
to equator and ecliptic arcs. Hence koti degree and 
nata degrees only should be used for ayana and 
aksa valana. 

Note : Correct method and meaning of terms 
has already been explained. 

Verses 70-77 : Diagram of eclipse— 

For making a parilekha (diagram), place is 
made plane like water level and a circle of 18 
angula semi diameter is drawn. East and west 
points are marked as explained earlier (in Triprasna 



540 Siddhanta Darpana 

dhikara) From these two points on circumference 
also two circles touching each other are drawn, 
each of 18 angula semi-diameter. In these two 
circles also, 4 points for cardinal directions and 4 
middle angles are marked. 

An east west line is drawn through centre of 
the two circles. A point is marked 1 angula north 
of north point of eastern circle and another point 
1 angula south of south point of western circle. 
When planet is in west kapala (west half of sky), 
a circle of 16/10 angula semi-diameter is drawn 
from southern point. When planet is in east kapala, 
same size circle is drawn from northern point. 

These arcs in the respective circles indicate 
kranti vrtta (ecliptic) 

Both arcs in the respective circles indicate 
kranti vrtta. In that signs of 12 rasis from mesa 
are given from west to east after making 12 equal 
parts. Centre of moon is kept in its correct rasi of 
kranti vrtta and around it, a partly eclipsed moon 
circle is formed. 

A line joining its two horns is drawn. The . 
line joining horns is equal to diameter of moon. 
With this diameter, circles are drawn at both 
external points of kranti vrtta. From this diagram 
moon will appear to be moving on kranti circle. 

In eastern circle, kranti vrtta is 328 yojana 

north (23 -° aksamsa of karka rekha) from equator 

which is line between east and west points of the 
circle. Kranti vrtta is actually a straight line, but 
appears curved due to drawing in a plane figure. 



Lunar Eclipse 



541 



Hence jya or kotijya are not needed in aksa or 
ayana valana. 

The curved shape kranti vrtta (and equator 
also) is perpendicular on all yamyottara (meridian) 
lines between two poles. Hence, on this kranti 
vrtta, distance from prime meridian (Ujjain or 
Greenwich) is desantara jya. Similarly, aksa jya is 
distance on north south line. 

Notes : This is like representation of earth in 
two touching circles in which karka rekha and 
makara rekha are north and south of equator. 



N 




Figure 19 
In figure -19 central circle is only for finding 
east west direction. East, west circles are of 18 
angula semi diameter in which all directions have 
been marked. P' is 1 angula north of N 1 , P 1 is 1 
angula south of 5. Kranti vrtta. Qi Q2, Q3, Q4 are 
drawn from these with 16/10 angula radius. This 
is only for explaination and not to the scale. 
However, this is a copy of school atlas map and 
reasonings about aksa valana and ayana valana on 
tn at basis are not correct. 



542 



Siddhanta Darpana 



Verses 78-79 : Effective shadow of earth. 

In moon's orbit, there is 5 kala less dark 
shadow (avatamasa or penumbra). On adding this, 
earth's shadow diameter increases by 10 kala. This 
semi dark shadow covers moon at other times also, 
then there is no eclipse but light of moon is 
dimmed. 

1/3 part of this semi shadow (penumbra) is 
very dark hence it almost merges with main 
shadow. Hence 1/3 of avatamasa or 10/3 kala is 
added to the earth's shadow to find the effective 
diameter of shadow. 

Notes: 

(1) M2 M3 penumbra in moon's orbit is formed 

F 




Figure - 20 

by direct tangents GB and transverse targent FB' 
(this will be very close to B). 

Z.FBG = 2R/r, where r is distance of sun 

R = radius of sun. 

Hence penumbra at distance ri of moon, 
making same angle. M2 BM3 is 

2Rri 
ri x L FBG = yojanas 

6 2 R ri 

x Kala 



85 



Lunar Eclipse 543 

6 72000 x 48,705 _ ,. ' , 

= — x ^7~^r~^r* = 32.5 kala 

85 76,08,294 

Thus the extent of lesser dark shadow is 
arbitrary. However, in penumbra, moon's light will 
be definitely lesser. 

As explained earlier, the effective increase of 
earth's shadow is by 2% or about 1 kala due to 
absorption by atmosphere. 

Verse 80-81 : Size of earth's shadow- It 

changes both due to sun distance and due to 
moon's distance, where its size is calculated. 

When sun is near mandocca, it is farthest 
from earth, hence shadow is bigger when gati is 
small and at 90° from nica, it reduces. Hence 1/28 
of ravigati phala is added to shadow or substracted 
from middle value. 

Moon's diameter is multiplied by 35 and 
divided by 13. In this, gati phala is substracted 
when positive. This gives true value of earth 
shadow. Method to find moon's diameter has 
already been stated. 

Note : True dimensions of shadow has already 
been stated based on true motions of sun and 
moon both in verse 27. This correction is based 
on arbitrary assumption of avatamasa' i.e. darker 
part of penumbra. 

Verses 82-83 : Calculation of true earth 
shadow-Due to relative rotation of sun around 
earth, earth shadow also rotates in same directon 
^th same speed, but always remains opposite. It 
covers moon according to its value in moon's orbit, 
here is difference of 1/20 parts due to variation 
m ^stance from sun. Due to varying distance of 



544 Siddhanta Darpana ; 

moon also its value changes. But this is very small 
compared to variation due to sun, hence it is 
neglected. 

Now method to calculate effective earth 
shadow is explained. From sun's diameter (72,000 
yojanas), its l/10th (7,200yojana) and earth diameter \ 
(1600yojana) are substracted. Remainder (63,200) is [ 
multiplied by mean moon distance (48,705 yojana) 
to get (3,07,81,56,000). This product is divided by I 
true distance of sun. The result substracted from 
earth diameter is the diameter of shadow in moon's 
orbit. This diameter multiplied by trijya (3438) and 
divided by true distance of moon gives angular 
diameter. 

Note : 'Avatamasa' (dark part of penumbra) • 
is 10/3 kala which is 1/12 of earth shadow (about 
40') Hence (sun diameter-earth diameter) is reduced X 
by 1/10 of sun diameter. Rest of the process is 
already explained in verse 26, whose diagram will] 
make it clear. 

Verses 84-86 : Colour of eclipse 

From the shadow of earth, 40 kala deducted! 
gives the value of andhatamasa (dark penumbra). \ 
(shadow is as calculated above). When sara of moon j 
is small, moon enters this dark penumbra and| 

. -J 

looks very dark. 

When lunar eclipse is very little, sky turns 
blue. In half eclipse, sky appears black. In morej 
then half eclipse, it looks red black. In total eclipse, 
moon becomes pale yellow due to its entry in 
earth's shadow. In solar eclipse, there is no change 
in colours; we seen only moon which is relatively 
dark. 



Lunar Eclipse 545 

Moon is always smaller than sun, even in 
angular diameter. Hence horns of sun are sharp 
in solar eclipse. But moon is cut by bigger circle 
of earth's shadow, so its horns are rounder in 
eclipse. 

Note : This is subjective description, hence 
no comments 

Verses 87-88 : Close 

Being dark in colour, shadow of earth is like 
rahu, in which moon enters at eclipse time and 
gives mantra siddhi to vaisnava and tantrikas. They 
may do good to us. 

Thus ends the eighth chapter describing lunar 
eclipse in detail in siddhanta darpana written for 
education of children and correspondance between 
theory and observation by Sri Candrasekhara born 
in famous royal family of Orissa. 

Eighth chapter ends. 



Chapter - 9 

SOLAR ECLIPSE 

Solar Eclipse 

Verse 1 - In last chapter, eclipse of moon and 
sun both have been discussed in a general way. 
For solar eclipse, in addition, it is necessary to 
calculate lambana and nati and bimba of moon 
(angular diameter) also is different for the purpose 
of solar eclipse. These three will be specially 
discussed in this chapter. 

Verse 2 : Reason of lambana and nati 

At the end of amavasya, rasi etc of moon and 
sun are same, even then they are seen in same 
direction only at the time of mid-day. On other 
times, they are not in the line passing through 
centre and surface point of observation. Why this 
happens for times other than mid day, will be 
described in this chapter. When sun and moon are 
in mid point of sky, their direction from centre 
and surface of earth is same. 

Verses 3-6 - Meaning of lambana and nati 

Sphuta ending time of amavasya calculated 
from sphuta moon and sun is called samaparva 
Kala'. This time after lambana correction is the 
middle time of grasa (eclipse) in solar eclipse. This 
is sphuta amanta time for the place. 

At amanta time calculated from earth's centre, 
the difference between directions of sun and moon 



Solar Eclipse 547 

is called lambana. This difference arising due to 
observation from earth's surface, and in east west 
direction is called lambana'. Its component in north 
south direction is called 'nati' or 'avanati'. When 
sun and moon are in mid sky, the line from earth 
centre to their centres passes through the surface 
point, hence there is no lambana or nati. 

When moon and vitribha lagna (lagna-90° on 
ecliptic) is same, there is no sphuta lambana, only 
nati is possible. When north kranti of vitribha lagna 
is same as (north) aksamsa of the place, then it 
has no nati also. 

When vitribha lagna' s north kranti is more 
than aksamsa of the place, moon (at vitribha lagna) 
has north nati. If north kranti of vitribha is less 
than north kranti of the place, or kranti is south, 
then moon has south nati. 

In amavasya (corrected with lambana), moon 
and sun have same rasi etc, hence nati in north 
south direction is easy to calculate. 

Verses 7-15 : Sphuta lambana by successive 
approximation - Instantaneous position of sun is 
found by method explained in sphutadhikara and 
from that, lagna of samaparva kala is calculated. 
By deducting 3 rasis (vitribha), again kranti is found 
for that. This kranti and aksamsa (direction of 
equator) being in different direction, difference is 
taken. They are added if they are in same direction. 

Result will be natamsa of vitribha lagna; On 
subs tr acting this from 90°, it gives unnatamsa. Jya 
of this unnatamsa is called sphuta drg-gati. 

Earth half diameter (800 yojana) assumed to 
be in sun or moon orbit, its angular diameter is 



54# Siddhanta Darpana 

found in kala. (For sun's orbit, it is divided by 
2213 and for moon's orbit multiplied by 6/85 
according to verse 15 of previous chapter). Result 
is called 'Kuchanna Kala'. This is equal to the 
parama nati of sun and moon. Difference of these 
two is the parama (maximum) nati in solar eclipse. 

„ . . Daily motion of sun 

Parama nab of sun = 

164 

_, ... Daily motion of moon 

Parama nati of moon = - — 

14 

Difference of parama nati of moon and sun 
in vikala is divided by difference of daily motions 
of moon and sun in kala. Result in danda etc. will 
be parama lambana time. 

Parama lambana time in danda etc. is multi- 

■ ■ 

plied by vitribha sanku of desired time and divided 
by trijya (3438). Result is antya of lambana. Jya of 
antya (in asu) is called para. 

Alternatively; sphuta drg gati (vitribha sanku) 
is multiplied by 100 and divided by 216. That will 
give the same para. 

Now jya of difference of vitribha lagna and 
sun is multiplied by para and divided by trijya. 
Result is lambana jya. Its arc in asu is sphuta 
lambana. If sun is west from vitribha lagna, this 
lambana time in asu is added to samaparva kala, 
otehrwise it is substracted. Result is sphuta 
samaparva kala. 

For this sphuta samaparva kala, we again 
calculate sphuta sun and vitribha lagna and 
lambana is calculated from their difference again. 
After repeated corrections, when there is no 



Solar Eclipse ^ 9 

difference between two samaparva kala, that is the 
correct larhbana. 

Notes - (1) Approximate use of this method 
has already been made in verse 4 of previous 
chapter to find possibility of solar eclipse. 

First we derive the equation of parama nati 
(already explained in appendix to triprasnadhikara). 




Figure 1 - Parallax ol moon 

C is centre of earth and M is moon. From a 
local place O, the moon's zenith distance is z' and 
z is zenith distance from centre of earth. If OC 
= p, radius of earth for the place and CM = r, 
distance of moon from earth centre, then in AOCM 



sin COM sin OMC 
But Sin L COM = Sin (180 °-z') = Sin z' 
L. OMC = L 71 OM -L OCM =z'-zo = p i.e. 
parallax. 

Thus Sin p = {plr) sin z. 

Maximum parallax P= ^ occurs when z=90° 

ie.Sin z = 1. 

This is parallax when moon is at horizon 

radius of earth 

Thus parama lambana P = -^. . , ^« "/ 

r Distance of moon 



550 Siddhanta Darpana 

This is angular radius of earth if it is kept in 
moon's orbit, hence it is called 'Kuchanna' ex- 
pressed in kala (minutes) i.e. ku = earth, channa 
= removed (to moon's orbit). Similarly parama 
lambana of sun is. earth's angular radius if it is 
viewed in sun's orbit. 

Alternative formula - For moon P in kala 

3438 6 
= radius, of earth x = — X radius of 

4o/UO o5 

earth 

(Verse 15 of previous chapter). 

But radius of earth = 800 yojana, moon's daily 
motion in kala is 790/35" which is slightly less than 
earth radius. Hence 

mon's daily motion 
P - if (2) 

Similarly parama lambana P' of sun is (mean 
value) 

n, ,. , , 3438 Earth radius kala 

F = radius of earth x — — — = 

76,08,294 2213 

(Result mentioned in verse 15 of previous 
chapter) 

= Sun daily moiton 

Earth radius 

x — — _ 

2213 x sun daily motion mean 
Sin daily motion x 800 Sun daily motion 

2213 x 59/8 = 164 ^ 

(2) Explaination of the terms : 

In figure 2, LNE is horizon and Z is zenith 
in celestial sphere. 

MVS is ecliptic and K its pole. 



Solar Eclipse 551 

M is meridian point of ecliptic and V is 
vitribha lagna, i.e., shortest distance from Z. Since 
ZV is perpendicular, it bisects the ecliptic above 
horizon, hence V is at 90° from horizon point L 
called lagna. Thus it is called vitribha or 3 rasi less 
(than lagna). 




Figure 2 - Explanation of drgjya etc 

ZA is perpendicular to MK, so sin Z A is drg 
gati of madhya lagna M (or smaller drggati). 

ZB is perpendicular to SK - so that R sin ZB 
is drggati of sun S (or larger drggati). 

ZS is zenith distance of S; R sin z is drg jya. 

ZM is zenith distance of M, R sin ZM is 
madhya jya. 

Distance from Z in direciton of ecliptic is thus 
drg gati. Distance from Z in direction perpendicular 
to* ecliptic is drkksepa. Thus drkksepa of M, V and 
S are AM, ZV, SB. Total distance from z is drgjya. 

(R sin ZS) 2 = (R sin ZB) 2 + (R sin SB) 2 
or drg jya 2 = drg gati 2 + drk ksepa 2 - - -( 4 ) 

(R sin SB) 2 = (R sin MA) 2 = (R sin ZM) 2 - (R Sin 

AZ) 2 

or drk ksepa 2 = drgjya of madhyalagna 2 
-drggati of madhyalagna 2 (5) 



552 Siddhdnta Darpana 

(3) Lambana antara of Sun and moon 



z 

K 




Figure 3 - Lambana of solar eclipse 
Portion ZVSB of figure z is repeated here as 
ZVMA. M is the common geocentric position of 
moon or sun at end of amavasya. 

S' and M' are apparent positions of sun and 
moon due to parallax, when viewed from surface. 
Thus MM' = p, MS' = p' * 

VM is ecliptic and K its pole. V is vitribha 
lagna, Z is zenith. 

M'D. and S'D' are perpendicular on ecliptic. 

M'B and S'B' are perpendicular on KM 
produced 

Arc MD or M'B is lambana i..e parallax of 
moon along ecliptic. 

Similarly arc MD' or S'B' is lambana of Sun. 
Arc D'D is lambana of solar eclipse or difference 
of lambanas of moon and sun. 

ZA is perpendicular to KM. Then from similar 
triangles MBM' and ZAM we have 

r» • /«* „v R sin ZA X R sin MM' 
R sin (BM') = 

R sin ZM 
But BM' = MD = R sin BM' approx 

or R sin MD = MD = — tS -^ — 

R Sin z 



Solar Eclipse 553 

= D?g S ati x P 3 * 3111 * 1 lambana P (6) 

Sin p 
as From (D P = ^ 

Similarly MD' = Drggati X P' (7) 
where P and P 1 are parama lambana of moon 
and sun. 

Thus DD' or lambanantara = MD-MD' 

= Drg gati x (P-F) from (6) and (7) 

= Drg gati x parama lambana antara (8) 
Parama lambana (antara) in time units is the 
time in covering that distance by moon. Relative 
speed of moon is moon gati - sun gati = m' kala 
Hence Parama lambana time 

Parama lambana kala , 

= day 

m' kala/day 

Parama lambana vikala 

= ghati 

m'kala B * 

Thus DD' in ghati 

= drg gati X parama lambana antara ghati 



(8a) 



(4) Vitribha Sanku and drggati - In figure 2 

ZV = nati of vitribha lagna 

= nati of equator - kranti ■ - - - (9) 

In figure (3), KV = KM=90° 

In similar triangles KZA and KVM 

RsinKZ _ R sin ZA 

RsinKV = RsinVM 

Rsin KZ x R sin VM 
or Drg gati R sin ZA = —=: 



554 Siddhdnta Darpana 

But R Sin KZ = R sin (90° - ZV) = vitribha 
sanku . 

R sin VM = ista sanku of sun or moon. 

tu t^_ Vitribha sanku X ista sanku 

Thus Drggan = ■- ( i 0a ) 



Radius 

vitribha sanku x Jya of vislesamsa 

Radius 



where VM = difference of sun and vitribha 
called vislesamsa 

From (8), laihbana 

Faram lambana x vitrib ha sanku 
= ^dh^s" xjyaof- 

vislesmsa (11) 

r» i , 56 x 60 

Parma lambana = — — danda 

731 

56 kala is moon's parama laihbana, sun 
lambana is negligible, it is converted to vikala, 731 
is difference of moon and sun gati 

56 x 60 x 360 
= — ^ asu = 1658 asu (taking 56/6/35 for 

56) 

Parama lambana 1658 100 

radius = 3438 = 216 ^ lla * 

Actually it comes 207, but after parallax in 
moon rise it is 216. 

(5) Summary of procedure - 

Natamsa of vitribha sanku is found from its 
kranti and aksamsa equation (9) 

'Para' is calculated from 100/216 x vitribha 
sanku (11a) 

Parama lambana x vitribha san ku 

radius (U) 

Then from equation (11) 



Solar Eclipse 555 

Lambana = Para x Jya of vislesamsa in asu 

Para x Jya o f vislesamsa . 
or Larhbanajya = — jj^ in kala 

For local place on surface, moon will be in 
same direction as sun before geocentric amanta 
when sun is in east. Because both move in east 
direction and in east half of sky moon appears 
further east due to parallax. In west sky, moon 
will be towards west, and it will reach sun's 
apparent positon towards east after lambana time. 

Lambana will change at new position, hence 
the procedure is repeated for further accuracy. 

Verse 16-22 : Accurate lambana in a single step. 

Now I tell the method to find accurate 
lambana in a single step. 

At samaparva kala, kotijya and bhuja jya of 
difference between sun and lagna is found. Square 
of (difference of para stated above and bhuja jya) 
and square of koti jya are added. Square root of 
sum is karna. Koti jya multiplied by para and 
divided by karna gives mean lambana time in asu. 
From this mean lambana, samaparva kala is 
corrected and drggati of that time is multiplied by 
madhyama lambana and divided by initial drggati 
(Here drggati means drg gati of tribhona lagna i.e. 
vitribha sanku). This is madhyama sphuta lambana, 
as stated by Bhaskara II. 

If this is more than madhyama lambana, their 
difference* in asu is squared, multiplied by madhya 
lambana. Result is added to madhya sphuta 
lambana. We take any of these - 1st sphuta lambana 
of Bhaskara or second sphuta lambana - multiply 



556 Siddhanta Darpana 

it by mean gati difference of sun and moon and 
divide by (first sphuta gati of moon - sphuta sun 
gati.) By this Bhaskariya lambana becomes more 
sphuta. With this value of sphuta lambana in ghap, 
parva kala is corrected as before. 

Alternatively, at sphuta parva kala, kotijya of 
difference of sphuta sun and lagna is multiplied 
by 276 x drggati and divided by trijya. This gives 
lambana in pala. 

This is multiplied by difference of mean gati 
of sun and moon and divided by difference of 
sphuta gati. This will give difference of sphuta and 
samaparva kala. 

Notes (1) Bhaskara formula - 



v 

— >C vP 

e /Sy \i 

/ G "V 

o^ A~7l 



Figure 4 - Bhaskara sphuta lambana 

E is earth, circle with centre E is sun orbit, 
circle with centre O is moon's orbit, deflected due 
to parallax. 

Vitribha lagna V and V of the orbits are in 
same direction from E, so that when sun is at V 
and moon at V, there is no lambana. 

Maximum lambana for this position of ecliptic 
is when distance from vitribha of sun and moon 
is 90°. This is OE. Directions OM and ES of sun 



Solar Eclipse 557 

from their vitribha is same, hence these lines are 
parallel and equal. In parallelogram ESMO, SM 
also is parallel and equal to OE. Thus SM is equal 
to parama lambana or 'para' in short. 

SM // VE, hence is perpendicular to horizontal 
lines at G and A. 

Now in similar triangles SDM and EMG, 
EG x SM 

SD = EM 

R sin (s ~ v) x para 
or R sin (lambana) = — (1) 

* 

Kar na EM ^___ 

= V M G 2 + EG 2 = V (SG - SM) 2 + EG 2 



= ^[R cos (S ~ V) - para] 2 + [R sin (s-V)] 2 •••• ( a ) 

drkksepa sanku x R si n 25-5fr- 
where para = — — - (b) 

We get maximum lambana when drkksepa 
sanku = R i.e. vitribha coincides with zenith and 
ecliptic is vertical. Then it is 1/14 of daily movement 
of moon which is 360° in angles. Thus max. 
lambana = 360714 = 25-5/7°. 

Putting values of (a) and (b) in (1) we get 
the formula. 

(2) Further corrections : Vitribha lagna is 90° 
from lagna by definition, hence V-L = 90 °, L = 
lagna. 

Then Sin (s ~ V) = Sin [(V-L) - (S-L)] = cos 
(S-L) and cos (S ~ V) = Sin (S-L) 

Further correction is based on sphuta gati 
difference of sun and moon as we had assumed 



555 Siddhanta Darpana 

average gati in formula (b) above. Larhbana angle 
= madhya lambana X madhya gati diff . 

= sphuta lambana time x sphuta gati diff. 
This ratio is basis for further correction. 
Verses 23-39 : Nati correction in sara 

After finding mid point of eclipse by methods 
described above, we have to find sphuta sthiti 
ardha in which sara of moon is to be corrected by 
nati. 

For this; last sphuta gati of moon is found 
for eclipse purpose as explained in chapter 6. That 
will be multiplied by sphuta lambana in ghati and 
divided by 60. Quotient will be added or sub- 
stracted in sphuta sun of samaparva kala as 
lambana correction. Pata of moon (rahu or ketu) 
is corrected with digamsa phala (1/10 of sun 
mandaphala - chanpter 6) and is substracted from 
sphuta moon. From this difference (candra-rahu), 
sara is calculated. 

Then from sphuta sun (sayana) of that time, 
lagna and vitribha lagna of sphuta parva kala is 
found and their kranti is calculated. 

South sara is added to aksamsa (where 
equator is towards south) and north sara sub- 
stracted to get saraksa. 

Saraksa and vitribha kranti are added, if in 
same direction or substracted for different direc- 
tions. Result is nata (north south distance of moon 
from zenith). Jya of this arc is natajya. This is also 
called versine of madhya lagna or udayajya. 

Madhya jya is multiplied by udaya jya and 
divided by dyujya. Square of quotient and square 



Solar Eclipse ^ 559 

of madhya jya are added. Square root of sum is 
drkksepa. 

Alternatively, vitribha lagna is assumed sun, 
and for that position kranti and cara are calculated. 

15 ghati + cara = dinardha and its difference 
with vibribha lagna is natasu. From this vitribha 
natasu; drkjya is found through utkrama jya, 
cheda, ista hrti, and vibribha sanku as explained 
in seventh chapter, (verses 45-51) 

Arc of this drgjya is vitribha natamsa. When 
vitribha kranti is north of local aksamsa, then 
natamsa is north, otherwise it is ' south. This 
natamsa and Sara of moon will be added, if in 
same direction, otherwise difference taken. Jya of 
the resulting arc is drkksepa. 

Thus there are two types of drkksepa. Both 
are separately multiplied by moon gati in kala and 
divided by trijya. Results are added when manda 
kendra of moon is in six rasis starting from karka 
(90° to 270°) or substracted for other six rasis. By 
this, drkksepa becomeis sphuta. 

When grasa is less than 1 kala or more than 
28 kala, then the second drkksepa is used which 
is corrected with vitribha natamsa. For grasa 
between 1 to 28 kala first drkksepa is used which 
is corrected with saraksa vitribha kranti (1/60 of 
total eclipse is one kala - when moon has sara, 
eclipse will be less than half and hence sara is 
used for correction). 

Difference of sun (21/38) and moon parma 
nati (56/28/13) i.e. (56/6/33) multiplied by drkksepa 
and divided by trijya gives sphuta nati. 



560 Siddhdnta Darpam 



Also sphuta nati = 



^ , , , Drkksepa 

Drkksepa - ' r 



61 

This is in direction of drkksepa. 

Sara of moon and sphuta nati are added if 
in same direction and difference is taken for 
opposite direction. Resulting direction will be 
direction of greater value of sara or nati. From 
that, grasa and sthiti ardha are calculated according 
to method stated in last chapter for lunar eclipse. 

When north sara of moon is more than the 
aksamsa of the place, aksamsa will be substracted 
from it. Difference will be north saraksa. When 
north kranti of vitribha is less than this saraksa 
but more than aksamsa, then aksamsa is sub- 
stracted from vitribha north kranti and result atided 
to saraksa gives north nata. 

Vitribha north kranti if less, is deducted from 
aksamsa and from remainder; saraksa is substracted 
to get south nata. If saraksa is more than | 
remainder, their difference will be north nata. 

Sum of vitribha south kranti and aksamsa if 
less than saraksa, their difference is north nata. 
These corrections are necessary for all places having 
more than 1° aksamsa. 

Thus in almost all places except equator 
region, natamsa is calculated from vitribha kranti 
corrected with saraksa; and drkksepa, madhya jya, 
nata in north south direction are calculated. 

Notes : (1) Moon's latitude from ecliptic 
depends upon its distance from pata (rahu or ketu). 
Its effective latitude for solar eclipse is latitude 
corrected for nati. 



Solar Eclipse 



561 



Now nati itself depdnds upon moon's distance 
from zenith towards south - consisting of two 
components - 

Distance of vitribha from zenith (it is only in 
north south direction - it is sum of aksamsa and 
kranti. 

Distance of moon from ecliptic - i.e. sara, 
Thus total distance in north south direction is 
algebraic sum of. 

Kranti of vitribha ± aksamsa of place ± sara- -(1) 

These are added if in same direction and 
substracted if in different direction. 

Total sara = sara±lambana - - (2) 

(2) Nati of moon : 

In figure 5, Z is 
zenith, V is central ecliptic 
point (Tribhona lagna), S 
the sun, S' apparent sun 
due to parallax, and S'A 
the perpendicular from S' 
on the ecliptic. Then from 
similar triangles SS'A and 

SZV, Figure 5 - Nati of moon 

S'A or sun's nati (approximately R sin S'A) 
_ RsinZVxR Sin SS' 

RsinSZ (3) 

Sun's drkksepa x R sin SS' 

*^m ,„^ ^^ „ 1 ■ ■ i — — — i.i -i ■ ' '■- * 

R sin SZ 
But R sin SS' 

parama nati of sun x R sin SZ 

" R 




- - (4) 



562 Siddhdnta Darpana 

_ Earth's semi diameter in yojana x R R sin SZ 

Sun's mean distance in yojanas R 

Hence, Sun's nati 

Sun's drkksepa x Earth's semi diameter in yojanas 

Sun's mean distance in yojanas 
Suns drkksepa x sun's true distance in yojana 

Sun's mean distance in yojanas 
Earth's semi diameter in yojanas 



x 



Sun's true distance in yojanas 
Sun's drkksepa x Sun's manda karna in minutes 

R 
Earth's semi diameter in yojanas 
Sun's true distance in yojanas 
or, Sun nati 
Sun's true drkksepa x Earth's semi diameter in yojana 

Sun's true distance in yojanas. 

Similarly moon's nati = 

Moon's true drkksepa x Earth's semi diameter in yojanas 

Mon's true distance in yojanas 

Alternate formulas 
From (3) and (4) 

Sphuta nati (difference of sphuta nati of moon 
and sun) 

Diff . of parama nati x drkksepa 

Radius K } 

as drkksepa of sun and moon is same when 
they have same longitude after lambana correction, 
giving the values — 

c k . -« aw 56/28/ 13 - 21/38 
Sphuta nati = drkksepa x 



Solar Eclipse 563 

\ / 

(3) Complete procedure for sthitiardha - 

(a) First of all, calculate the time of geocentric 
conjunction (ganitagata or karanagata darsanta or 
amanta). Then calculate the lambana for that time 
and treating it as lambana for the time of apparent 
conjunction, obtain the time of apparent conjunc- 
tion by the formula - 

Time of apparent conjunction = Time of 
geocentric conjunction ± Lambana for the time of 
apparent conjunction (1) 

+ or - sign being taken according as the, 
conjunction occurs to the west or east of the central 
ecliptic point. Next, calculate the lambana for the 
time of apparent conjunction obtained and then 
again apparent conjunction is calculated from 
formula (1). 

For the time of this second apparent conjunc- 
tion, lambana is calculated and again aparent 
conjunction is calculated (third) by formula (1). 

This process is repeated till lambana for the 
time of apparent conjunction is fixed. Applying 
this lambana in formula (1) we get the correct time 
of apparent conjunction. This is the time of spasta 
darsanta or spasta amanta, and also the time of 
middle of the eclipse. 

(b) Sparsika and mauksika sthiti ardhas - 
Calculate the semi diameters of the sun and moon 
and also moon's true latitude corrected for nati as 
explained in notes (1) and (2), for the time of 
apparent conjunction. This is almost equal to 



564 Siddhanta Darpana 

moon's latitude at first contact time pi. If S and 
M are semi diameters of Sun and moon, d is 
difference between true daily motion of moon and 
sun in degrees - 

sparsika sthityardha = — * -f ^- ghatis - - (2) 

In practice, one uses the semi diameters of 
the sun and moon for the time of apparent 
conjunction, because, for the time of first contact, 
there is negligible change. 

Therefore, time of first contact 

= Time of apparent conjunction - sparsika 
sthityardha (3) 

Next, calculate the moon's true latitude for 
the time of first contact thus obtained; and then 
find the sparsika sthityardha by formula (2), then 
time of first contact by formula (3). 

Then calculate the moon's true latitude for 
the time of first contact (2nd value), then calculate 
the sparsika sthityardha by formula (2) and time 
of first contact by formula (3) again. 

Repeat this process unitil the sparsika 
sthityardha and the time of the first contact are 
fixed. 

The sthityardhas and vimardardhas which are 
thus obtained are called madhyama (or mean), 
because they are still uncorrected for lambana. 

(c) Lambana for times of apparent first contact 
and separation-Calculate the lambana for the time 
of first contact obtained above and treating it as 
the lambana for the time of apparent first contact, 
obtain the time of apparent first contact by the 
formula — 



Solar Eclipse 5*5 

Time of apparent first contact = time of first 
contact ± lambana for the time of apparent first 
contact - - - - - (4) 

+ or - sign being taken according as the first 
contact takes place to the west or east of the central 
ecliptic point. 

For the time of apparent first contact, thus 
obtained, calculate the lambana afresh and applying 
it in formula (4), obtain the time of first contact 
again. 

Repeat this process until the lambana for the 
time of apparent first contact is fixed. 

Similarly, find the lambanas for the times of 
apparent separation, immersion and emersion 

(d) Sparsika and mauksika sthityardhas, cor- 
rected for lambana — 

The madhyama sparsika and madhyama 
mauksika sthityardhas corrected for lambana, are 
called true (sphuta) sparsika and sphuta mauksika 
sthityardhas. They are obtained by the formula. 

True sparsika sthityardha = time of apparent 
conjunction - time of apparent first contact 

True mauksika sthityardha = Time of apparent 
separation - time of apparent conjunction. 

Similarly, 

True sparsika vimardardha = Time of apparent 
conjunction - time of apparent immersion 

True mauksika vimardardha = Time of ap- 
parent emersion - time of apparent conjunction 

Verses 40-42 : More accurate value of moon 
diameter (bimba) - Bimba (angular diameter) of 



566 Siddhdnta Darpana 

sun and moon is calculated according to method 
given in previous chapter on candra grahana. Now 
method is being given to make it more accurate. 
This has not been told by any earlier scholar 
(acarya). 

If manda kendra of moon is in six rasis 
starting with karka, its koti phala is substracted 
from trijya, otherwise they are added. Square of 
the result is added to square of manda bhujaphala. 
Square root of sum is substracted from twice the 
trijya. By remainder, square of trijya is divided. 
Result will be manda karna in lipta (i.e. kala or 
minute of arc). Mean birhba kala of candra (31/20) 
is multiplied by trijya (3438) and product (107724) 
is divided by manda karna. It will give sphuta 
bimbamana of moon. 

Notes (1) If R and r are radius of main circle 
and manda paridhi, then 

Koti of karna = R + r cos 

when is manda Kendra 

Bhuja of karna = r Sin 6 

Hence, karna K is given by 

K 2 = (R + r cos Of + (r sin 0) 2 (1) 

This is the correct formula. However, in place 
of bhuja phala or koti phala we take the lower 
value 

Bhujaphala = r sin x R/K 

Similarly, kotiphala, r cos also in reduced 
in same ratio. 

Thus we take 



K! 2 = 



R + r cos $ x — 

\ / 



2 '' rsinfl x R x2 

K 



Solar Eclipse 



567 



= R 2 + ^ (cos 6 + Sin 6) 



+ 



^ 2 
K 



(cos 2 6 + Sin 2 0) 



rR 
or Ki = R + " (cos0 + Sin ^) approx. (2) 

2 R-Ki = R - — (cos 6 + Sin ) 

XV 

R 2 R 2 



2R - Ki rR 

R - — (cos + sin 6) 
is. 



R 



1 - £ (cos + sin 6) 

= R [1 + -^ (cos 6 + sin (9)] = K from (2) 

(2) Mean bimba X mean distance (trijya) 

= True birhba x true distance (manda karna) 

= Diameter in length units. 

Verses 43-45 : Methods for calculating 
tamomana - 

(1) At the time of sphuta amanta time, we 
find sanku and drgjya from spasta sun. Parama 
lambana (56/28) is substracted from sanku. Squares 
of remainder and drgjya are added and of the 
sum, square root is taken. This will be tama 
karna (chaya karna). Sphuta candra birhba is 
multiplied by trijya and divided by tama-karna 



568 Siddhanta Darpana 

This gives tamomana or grahaka (eclipser) value 
in solar eclipse. 

(2) Alternatively, 1/60 of sphuta candra bimba 
is multiplied by sanku of sphuta parva time and 
divided by trijya. Quotient is added to sphuta 
candra bimba to get tamomana. 

(3) Due to hard labour involved in calculating 
tamo bimba through sanku etc., I have found an 
easy method also for this. Unnata kala in ghati at 
the time of sphuta parva kala is multiplied by 2, 
the product in vikala is added to sphuta candra 
bimba in kala etc. 

Notes : (1) In previous verses sphuta candra 
bimba has been calculated for its variation in 
distance from earth's centre. However, due to 
parallax in observing moon from surface, its angle 
from vertical is increased, but distance is decreased. 
Though we correct the angle difference, the 
distance difference still remains. Since moon is seen 
at a nearer distance due to parallax, its effective 
angular diameter will appear increased. We have 
to calculate the increased bimba mana. Here 
tamo-mana is not the value of shadow, because 
shadow is not the cause of solar eclipse. Moon 
disc itself appears dark compared to sun and is 
called tama. 

(2) Derivation of formula - Figure 4 after verse 
22 may be referred to For clarity, a smaller figure 
is made here (figure 5.) OZ is vertical and ZSH 
the great circle from Z through S, centre of sun 
and Moon. OH is horizon line. Sun and moon are 
at same place on samaparva kala, but figure at 
amanta time is shown when M is separate due to 



569 



Solar Eclipse 

parallax, so that distance difference is shown. 
iSOZ = Z is distance from vertical. SP is sanku 
= R cos Z, and SN = R sin Z is drgjya. 




Figure 5a - Tamomana increase in bimba due to parallax 

Due to parallax, moon is lowered to M' where 
SM' is equal to parama lambana of moon (as in 
figure 5a). 

SM' = P. On celestial sphere moon is seen at 
M in that direction. SM is small and this arc and 
straight line are almost same. 

In right angled triangle SMM',Z SMM' = 90°, 
L SM'M = z' when ZOM = z'. At sphuta samaparva 
kala z = z'. 

Thus lambana SM = SM' sin z' = P Sin z' 
which is according to the formula for lambana. It 
comfirms that apparent height of moon is lowered 
by distance SM' = parama lambana. 

Apparent distance from surface is OM' = 
tamo-karna of moon 

OM' 2 = OP 2 + PM' 2 

= NS 2 + (SP-SM') 2 

or tamokarna 

= V drgjya 2 + (Sanku - parama lambana) 2 " 0) 

Sphuta bimba of moon has been calculated 
for the distance of radius OM from earth's centre. 



/ Trijya _ x 
Tama Kama 



parts of 



^ Siddhdnta Darpana 

Apparent bimba at M' is bigger, which is 
tamomana. 

Hence, linear diameter being same 
linear diameter = tamomana X OM' 
= sphuta bimba X OM 

Sphuta bimba x Triiya 

or, tamo mana = — — — ^— o) 

Tama karna v ' 

(3) Alternate formula - 

Increase in sphuta bimba = 

bimba 

= OM - OM' _ M'M _ P cos z / 

OM' " OM' " R - P Cos z' (3) 

p. 

when z = O, at Z, increase is maximum = 

R- P 

Absolute in crease is P cos z' 

Then fractional increase in sphuta bimba 

_ P Rcosz' 

- R _ p ^ — / as P cos z' « P 

56/28 Sanku J_ Sanku 

3438 - 56/28 Trijya " 60 Trijya (4) 

In 15 ghati unnata kala, increase in moon 
bimba is 1/60 of sphuta bimba = J/60 x 30 kala 
approximately, when moon is at Z. 

1 30 
Hence in 1 ghati increase is — X — kala 

60 15 

= 2 vikala approximately 

Thus for each ghati unnata kala, sphuta bimba 
increases by about 2 vikala. 

Verses 46-47 : Hara of solar eclipse. 

Sphuta candra gati is multiplied by 1/60 of 
sphuta samku of sun and divided by trijya. In 



571 
Solar Eclipse 

auotient, final sphuta gati of moon is added. Sum 
substracted from sun gati will be hara at the time 
of eclipse (mid time). 

Unnata kala in ghati is reduced by its 1/8, 
remaining is assumed as kala and added to final 
sphuta gati of moon and sphuta gati of sun is 
substracted. Result is hara of sparsa and moksa 

time. 

Notes : (1) Hara means multiplier; here the 
purpose of this multiplier is not mentioned. 
However, in verses 46-50 of previous chapter on 
lunar eclipse, hara is used for calculating amount 
of grasa (magnitude of eclipse) at desired time. 
Hara in that context is difference of moon's speed 
and sun's speed. For solar eclipse this needs 
accurate calculation and correction for lambana. 

Hara = Candragati - surya gati (1) 

In this, variation due to parallax is only in 
candragati as the parallax of sun is negligible. 

linear d iameter 

Bimba = — ttt 

true distance 

linear motion 
^ true distance 

Thus bimba and gati of moon both increase 
in same proportion due to apparent decrease in 
distance due to lambana or parallax. 

Thus according to first alternative formula in 
note (3) of previous verse, equation (4) is 
Proportional increase in candragati 
1 sphuta sanku ,~\ 

60 Trijya 



Siddhanta Darpana 

This correction put in equation (1) gives the 
first formula for lambana corrected hara 

(2) Hara for sparsa or moksa time - 

At 15 ghati unnata kala the increase in 
candragati from (2) is 1/60 part of its gati, . when 
moon is at Z approximately. 

This increase is 79035/60 = 13-1/6 kala ap- 
proximately. Hence, proportionate increase of each 
ghati in moon gati is 

13 1 

6 __ 11 ^ 1 

15 ~ 1 ~6X15~ 1 "8 kala apprOX - 

Thus ghati is reduced by its 1/8 and remaining 
part taken as kala is the increase in daily motion 
of moon due to parallax. For this, unnata ghati of 
sparsa or moksa time is taken. 

Verses 48-49 : Difference in solar eclipse at 
each place. 

In lunar eclipse, shadow of earth and moon 
- both are at same place (in moon's orbit), hence 
grasa is same at all places, because there is no 
parallax. But in solar eclipse, chadya sun and 
chadaka moon are very far from each other. Only 
at a particular place, they may be in one line, but 
at other place they will be seen in different direction 
due to lambana (or parallax). Thus solar eclipse 
has different magnitudes for different places). Even 
due to a small difference in east west or north 
south direction, there will be difference in total 
eclipse, annular or partial eclipse. Hence, they are 
to be calculated separately for each place. 

Notes : Location of the point of observation 
is only reason for solar eclipse, other wise they 



Solar Eclipse 573 

are vastly far from each other. This has been 
explained in beginning of previous chapter and 
while calculation of solar eclipse also. 

Solar eclipse is seen in a very small circle cut 
in moon's shadow cone by earth's surface. In north 
south direction from that circle, eclipse will become 
partial and then non existant. 

Due to relative motion of moon towards east 
the shadow circle on earth's surface moves from 
west to east and finally leaves. Thus the eclipse is 
earlier in west and later in eastern places on the 
strip of earth surface. Thus due to east west 
difference of places, eclipse times and grasa times 
will be different (according to standard time also). 

When tip of shadow cone is about to leave 
earth surface, before and after the strip, when circle 
on surface is of zero radius, extended shadow cone 
touches the surface. Then annular eclipse is seen 
at those places. 

Verses 50-53 : Madhya sphuta sthiti kala 

According to rules explained in candra 
grahana chapter, we calculate the sphuta sara, half 
sum of bimba. From hara of grahana time we 
calculate the sthiti ardha and marda ardha in ghati. 
By adding or substracting this from samaparva kala, 
we get times of sparsa, moksa, sammflana and 
unmilana. 

Then current lagna and vitribha lagna is found 
and lambana in east west direction is calculated. 
Sparsa and moksa times are corrected with this 
lambana. For these sparsa times etc, we calculate 
the lambana again and second value of sthitiardha 
and sparsa kala is found. For second values of 



574 Siddhanta Darpana 

sparsa and moksa times, lambana is again calcu- 
lated and from that we get third value of sparsa 
or moksa. After repeated process, when there is 
no difference in successive values, we get the true 
values. 

Verses 54-56 : Sphuta sthiti kala by sara 
correction From difference of sphuta parva kala 
and these times of sparsa etc., we get the values 
of both sthiti ardha and marda ardha in ghati etc. 
Alternatively, we find the sphuta sara by single 
step method (verse 45 of previous chapter), and 
new values are found. From their ratio, sara for 
sparsa and moksa time is found. One difference 
is + ve and other is negative. Both changed by 
half the sum give the sara of sparsa, moksa time. 

From this sara, second value of second sthiti 
ardha is found. From that we find sara for sparsa, 
moksa and middle time sara. Then we find the 
difference of middle sara with the sara of sparsa 
and moksa times. By proportionate difference we 
again find sphuta sara ardha. After repeated 
process sthiti ardha becomes spasta. 

Notes : (1) Correction of sthti ardha for 
lambana by repeated process has already been 
explained after verse 53 and in notes after verse 
39. 

(2) Suppose the sara at middle time be L and 
sparsa time sara is U. By single step method, the 
spasta sara is L'. Thus difference of sara is L'-l, in 
single step method and L-li in repeated method. 
Thus the difference of single' step method is to be 
changed by (L-li)/(Li-li) for correct difference. Thus 
we get accurate sara by one step method. If sparsa 



■3 



575 
Solar Eclipse 

time sara is less than middle time sara, moksa time 
Sara will be more. 

Verse 57 : Method for small sthiti ardha We 
take the difference of sthiti ardha after 1st lambana 
correction and the sthiti ardha before that correction 
(initial value). Square of difference in pala is 
divided by initial sthiti ardha. Result is added to 
sthti ardha obtained initially. 

This process is done only for sthiti ardha less 
than 1 danda. From new values we get correct 
sparsa time etc. 

Note : Let the sparsa times counted from 
middle eclipse time be t , ti and t 2 before sara 
correction and after first and second sara correc- 
tions. For small sthiti ardha, second corrected time 
t 2 will be almost correct time. Change in sthiti 
ardha after 1st correction is 



/ 



ti - to = to 



1-t 

to 



same proportion 



\ 



It is assumed that sthiti ardha will change in 

/ \ 

h 

to 

(ti - to) 2 



- 1 



t2"to = to 



to 



\ 



in next step also. 



2 - 



Thus the correction is obtained by dividing 
square of difference of initial and first corrected 
sthiti ardha by initial sthiti ardha. 

Same process can be used for moksa time 
also. Proportional decrease or increase can be 
assumed only for small sthiti times. 



576 Siddhanta Darpana 

Verses 58-60 - Single step method for sphuta 
sthiti time. We obtain sphuta sara for sparsa or 
moksa times after adding or substracting madhya 
sthiti ardha from lambana corrected amanta. If this 
sara is more than sum of semi diameter of the 
birhba; or equal to it, then madhya sthiti ardha is 
multiplied by grasa kala and made half. It is divided 
by difference of parva kala sara and sara at sparsa 
or moksa time (expressed in kala). By this, moksa 
and sthiti ardha are found in a single step only. 
From sthiti ardha times obtained, the corrected 
middle time gives sphuta lambana in one step only. 
Then sphuta sara will be found for lambana 
corrected sparsa and moksa times in one step only. 

Note (1) Grasa kala is amount of grasa 
expressed as ratio of diameter of eclipsed planet, 
out of total kala of 60. Thus 

sum of semi diameters - sara 
8 Diameter of eclipsed graha 

When sara is more than semi diameter sum, 
then the planet will not be eclipsed and eclipse 
time will be shortened. 

Average value of sara between madhya kala 
and sparsa time is taken. When grasa is small, its 
value nearer to middle time is taken, as the real 
sthiti ardha itself is shortened. 

Verses 61-62 : Annular eclipse 

In solar eclipse when birhba of sun is more 
than tamo-bimba (apparent birhba of moon in- 
creased for parallax, then eclipse will be annular 
(valaya grasa). Then, from sum of semi diameters, 
diameter of moon is substracted. From square of 



A 



;i 



Solar Eclipse 577 

the difference, square of sphuta sara is substracted. 
From square root of this difference, we find sthiti 
ardha etc. in pala as per method described in lunar 
eclipse chapter. This sthiti ardha pala is corrected 
for larhbana and on adding or substracting from 
samaparva kala, we get beginning and end times 
of valaya grasa. 

Notes : This method is same as that of total 
eclipse time in which difference of semi diameter 
is taken. In this case, we get valaya grasa instead 
of total eclipse, because moon bimba is smaller. 

Verses 63-64 - Reason for extra methods 

Brahmagupta (son of Jisnugupta) had ob- 
served errors in the calculation of eclipse durations, 
hence in his Brahma-sphuta-siddhanta, stated at 
the end of tithi chapter, corrections for nadi (ayana 
drk karma), bhuja of nata, its jya etc. 

The method described by Bhaskaracarya in his 
Siddhanta Siromani also doesn't give correct eclipse 
duration. Hence, on the difficult topic of solar 
eclipse, I have stated many more things. 

Notes : Already many new improvements 
have been described to get more correct values of 
moon bimba etc. Now entirely new methods are 
being described for correct duration of eclipse. After 
that, modern methods will be described, as 
comments. 

Verses 65-72 t Eclipse duration through yasti 

- After calculating surya grahana by above rule, 
we multiply the sphuta Sara at the time of sparsa, 
middle and moksa, separately by the lagna kranti 



578 Siddhanta Darpana 

jya of their respective times to give yasti for the 
three times. 

The three yastis are converted to para (1/60 
vikala) and divided by hara (candra gati - surya 
gati) for the time of sparsa etc. When lagna kranti 
and sphuta sara are in different direction, this 
result in pala etc is added to time of sparsa etc 
otherwise substracted. Then true sparsa, madhya 
and moksa times are obtained. 

If this time is more than previous time (i.e. 
yasti -s- hara is added for different directions of 
lagna kranti), then it is the true time for sparsa 
etc. If new time is less then previous, it is 
multiplied by its lambana jya and divided by 'para' 
(stated arlier). Result is added to sparsa time etc., 
when sun is west from vitribha lagna, otherwise, 
it is substracted. This will give true times of sparsa, 
madhya and moksa. Madhya time will again be 
corrected with sphuta lambana to get correct value. 

Then squares of mid time sara and yasti arej 
added and square root of the sum is sphuta 
madhya kala sara. Then from the sara, sthiti ardha 
for sparsa and moksa are found. They arej 
separately multiplied by sphuta lagna dyujya for: 
madhya kala and divided by trijya. 

When sara of sparsa and moksa is in samei 
direction, first result is substracted from sparsa^ 
time and second result is added to moksa time. 
When the two sara are in different direction, 
reverse is done. 

The sparsa and moksa times are corrected for 
their lambanas to get true values. But sthityardha 



579 
Solar Eclipse 

is multiplied by dyujya of madhya kala and divided 

bv triiya. 

Lambana for parvanta is found from true sun 
of that time. At the time of sparsa and moksa, 
°ambana is calculated from position of moon at that 

time. 




Figure 6 - Sara correction through yasti 



Notes : To explain yasti, figure 10 after 
Triprasnadhikara verse 37 is "produced here. 
NZSZ'' is yamyottara vrtta, NS is horizontal line, 
ECE' is diameter of equator. 

QQ' is diameter of diurnal circle of sun and 
LI/ is diurnal circle of moon further removed from 
equator as kranti and sara are in opposite direction. 
These three circles are bisected by perpendicular 
PP' through poles - which is diameter of 

unmandala. 

BQ = Dyujya = semidiameter of diurnal circle 

= R cos <5 = corresponding to equator half 
day CE = 6 hours 

BD' = Kujya = Extra length of half day on 
diurnal circle = BC tan<l> 

H 

= R sin d tan 0> 



tjgQ Siddhanta Darpana 

CD = Carajya = Extra length of half day on 
equator in asu 

BD' 

= = R tan d tan# 

coso 

Now B is the position of sun when it has 
risen on equator. BD = height of sun at that time 
i.e. unmandala sanku. 

Height of planet above B is called yasti. 

Now A' is the joint position of sun and moon 
on ecliptic, A its position on equator corresponding 
to arc CA in asu. Let CA = K 

Height of A = CA cos <$> , where <S> is latitude) 
= K cos ® 

Height of A' above B i..e yasti of A' is 

= A'B cos O = K cos O cos d 

Its rate of increase with respect to angular 
distance from equator is 

- K cos O sin d 

Hence for change in distance corresponding 
to §ara s of moon, 

Increase in yasti = s k cos <I> Sin <5 

s k cos <p sin d 
Proportionate increase * ^ cos< p 

= s sin d ---.- r - --(1) 

This yasti is the proportionate increase in time 
units of yasti and not the ista yasti meant in chapter 

Thus increase in yasti is equivalent to decrease 
in lambana, hence moon will reach the sparsa time 
after corresponding interval. Thus increase in 
sparsa time = yasti/hara or para where hara is 



Solar Eclipse 581 

Relative speed of moon. When yasti is in vikala/60 
and hara is in kala/day, the result is in day X 60 
X 60 = in palas. Similar addition is to be made for 
J$he times of madhya and moksa also. When sara 
fis in same direction as kranti, substraction is to be 
^jiade. 

When times are to be deducted they are 
Changed in ratio (lambana jya / sama mandala 
Iganku), because lambana jya is in time units. 

I Yasti is correction in sara of all times, hence 
leverage mid time sara is obtained by 

|§ara 2 +yasti 2 .) 1/2 

tlferses 73-82 : Miscellaneous corrections 

I If among sparsika and mauksika saras, one is 

Nqual to middle time sara and other bigger, then 
llhere is a special method. 

I Ecliptic times are found by above methods 

land the sphuta sara of sparsa, madhya and moksa 
[Sime are multiplied by the kranti jya of lagna of 
Jtheir times and divided by trijya. When sara of 
j sparsa and moksa time are in same direction, these 
fresults are substracted from their sara> added if in 
| different directions. Result is multiplied by dif- 
|ference of sara and divided by 36. We get yasti in 
§!ipta. 

I This is multiplied by jya of distance between 
f&un and vitribha lagna and divided by trijya (3438), 
to get the third yasti. This third yasti in para is 
divided by sthiti ardha for sparsa etc and the result 
m pala etc is added to the times of sparsa etc. 
*vhen kranti and sara are in different directions, 
otherwise substracted. Thus we get the true times 
of sparsa, madhya and moksa. Madhya kala is 



I 



15 



<$i 






B&\. 



coo Siddhanta Darpana 

again corrected with sphuta lambana to get correct 

value. 

Then madhya kala sara and madhya yasti - 
both are squared, added and of the sum square 
root is taken. With this sphuta madhya kala sara, 
we calculate the sthiti ardha for moksa and sparsa 

limes. 

These are separately multiplied by dyujya of 
madhya kala lagna and divided by trijya. First 
result is substracted from sthiti ardha of sparsa 
and second is added to moksa sthiti ardha. Then 
both are corrected for their lambanas. 

When difference between spasta sara of 
madhya kala, and sum of semi diameters of birhba 
is more than 3 kala and kranti of sun is more than 
lagna kranti then surya grahana is calculated 
according to this method. 

If madhya sara is less than both the saras at 
sparsa and moksa time, more than both or equal 
to both, then first method should be used. 

Notes : (1) Kranti of sun is between the kranti 
of lagna and kranti of vitribha lagna, hence it to 
aproximated by either of them, which are at 90 
from each other. No earlier astronomer had used 
kranti of lagna from whiich eclipse time can be 
calculated through yasti difference. Yasti differernce 
is same as difference of sanku. Both methods give 
same errors. In calculation with yasti one tune 
method has been used for calculating sthiti ardnas 
with sphuta sara corrected for yasti. 

(2) This method of yasti and previous methods 
are almost same. When grasa is 3 kala or more, 



Solar Eclipse 583 

(difference of sara and sum of semi diameters), 
then the approximate distance between sparsa and 
moksa places will be (sun birhba + 3 kala) = 36 
fcala aproximately. Hence sara difference is divided 
by 36 and resulting yasti is added to middle time 
jara. Approximately same will be added to other 
Saras also. 

Verses 83-85 Only that grahana (eclipse) is 
meaningful, which is seen from local place. No 
auspicious functions are needed for the grahana 
pot seen at a place. Thus lunar eclipse in day time 
:§t. solar eclipse in night time are not considered 
|is grahana for that place. 

But even at the time of part solar eclipse in 
liay time or part lunar eclipse in night should be 
Observed according to smrtis. Bath, charities etc 
should be done; cooking sleeping etc are 
prohibited. 

, As in lunar eclipse, in solar eclipse also grasa 
from time and time from grasa is calculated. Simlar 
method is used for aksa and ayana valana. 

Note (1) Amount of grasa and time in solar 
-eclipse. 

Let T be the Indian standard time of conjunc- 
tion in longitude, p is latitude of the moon, Pthe 
hourly change in latitude (north latitude and 
feotion towards the north being considered posi- 
tive), -M is excess of hourly motion of moon in 
tengitude over that of sun. 

L is angular radius of moon, S angular radius 
<?f Sun. Then at anytime t hours after conjunction, 
Ihe distance between the sun and moon's longitude 



■' ■'.'*' 
■•''■' ..^ 






mi 



584 Siddhdnta Darpana 

is Mt and the moon's latitude is (p + Pt). So the 
distance between their centres is [M 2 t 2 + (p +. Pt) 2 ] 1/2 

The eclipse begins or ends, when their rims 
appear to touch. This can happen, even if the 
distance between them is greater than L+S, for the 
moon's parallax may push it towards the sun. The 
maximum of this effect is II-H' (= II); II being the 
equatorial horizontal parallax of the moon, II' of 
sun which is negligible. 

Thus the rims can appear to touch when the 
distance between the centres is II + L+S (=d) at 
the most. Then [M¥ + (p+Pt) 2 ] = d 2 gives the 
times of the beginning and end of the general 
eclipse. Solving for t, we get 



-pP - 

* M 2 + p 2 + 



» 2 P 2 d 2 -p : 



M 2 + P 2 M 2 + P 2 



Vi 



In this, the upper sign (-) is taken for the 
beginning, and lower for the end. T+t is the 1ST 
of the beginning or the end. 

At any given place, the eclipse begins or ends 
when the rims appear to touch at that place, i.e. 
when the apparent distance between centres is L+S. 
Now at any time T near the times of conjunction 
in longitude, let the apparent distance in longitude 
between the centres be m, the apparent excess of 
moon's hourly motion in longitude over the sun 
be M, apparent difference in latitude p, apparent 
excess of moon's hourly motion in latitude over 
that of sun be P, the sum of angular radii of sun 
and moon be d, and its variation per hour D. By 
apparent is meant here '(as affected by parallax)'. 



Solar Eclipse 5S5 

Apparent m = real m + II cos A. Cos B (1 + 
II cos A Sin B) 

Apparent p = (real p+II Sin A) (1+H cos A 

Sin B) 

Apparent (L+S) = S+L (l+II cos A. Sin B) 
where A is the zenith distance of vitribha 
lagna given by 

Sin A = sin co cos <p sin v - cos to sm<p 
and B is (lagna - moon's longitude) given by 

B = Tan ' l [tan 1/2 (90° +v) cos 1/2 (90° + <p 
-w) / cos 1/2 (90°+ c/> +w)] 

♦Tan" 1 [tan 1/2 (90°+v) Sin 1/2 (90°+ 0-w) Sin 
1/2 (90°+ <p +w)] 

where ^ = latitude of the place 

to = obliquity of ecliptic (parama kranti) 

and v = sidereal time in degrees at the moment 
given by v = 97° 30' + east longitude of place in 
degrees from Greenwich + mean longitude of sun 
+ 1ST at that moment in degrees. 

For strict accuracy, the geocentric latitude and 
horizontal parallax at that latitude should be used. 

If T is the time for which we have found m, 
p and d, the apparent distance between the centres 
of the sun and the moon at any time t hours after 
T is 

[m+Mt) 2 + (p+Pt) 2 ] 1/2 

When this time is equal to d+Dt, the eclipse 
begins or ends. Thus eclipse begins or ends at 



586 Siddhanta Darpana 

dD-mM-pP 



T + 



+ 



M 2 + P 2 

i Vi 



(mM + pP - dD) 2 d 2 - p 2 - m 2 
(M 2 + P 2 ) 2 + M 2 + p 2 



The middle of the eclipse i.e. the maximum 

dD - mM - pP 
eclipse occus at T + — ; — . 

r M 2 + P 2 

The total eclipse begins or ends, when the 
rims apparently touch, the sun being within the 
moon. The distance between them at such time is 
(L-S), so by substituting for d in the above formula 
another d equal to (L-S), we can find the times of 
the beginning and end of the total phase. 

S may be greater than L, so that moon may 
be immersed in the sun, leaving a circle of light 
all around. This is called annular eclipse. Beginning 
or end of the annular eclipse is got by making D 
= S-L. 

(2) Bessel's method - for calculating solar 
eclipses - Bessel's method for calculating the 
circumstances of a solar eclipse as seen from a 
given place on the surface of earth consists in 
choosing a suitable system of axes, finding coor- 
dinates of the observer with respect to these axes 
and putting down in terms of these coordinates, 
the condition that the observer lies on the boundary 
of the penumbral cone at the beginning or end of 
the eclipse. All variable quantities in this condition 
are written in the form x +x't, where Xq is the 
value of the variable quantity at t = o and x 1 is 
the rate of change of the variable quantity. The 
origin of time is chosen near the middle of the 
eclipse so that t is small. The condition now 



Solar Eclipse 587 

becomes a quadratic equation in t, solving which 
we know the beginning and the end of the eclipse. 

Besselian elements - Through the centre E of 
the earth draw a line paralled to the line joining 
the centres S, M of the sun and moon. Call this 
Z axis, its positive direction being on the side on 
which sun and moon are situated. 

Choose the y axis to lie in the plane 
determined by the z-axis and the axis EN of the 
earth, the positive direction of y axis making an 
acute angle with EN. Finally choose the x-axis to 
be perpendicular to the axis of y and z, its positive 
direction being towards the point of equator, which 
the earth's rotation is carrying from the positive 
side to the negative side. 




East 



Figure7 - Bessalian elements for solar eclipoe 

The plane z = O is called the fundamental 
plane. 

These axes are not fixed with with respect to 
the surface of the earth. Therefore, the coordinates 
of a point on the surface of earth keep changing. 

Certain quantities need to be calculated first 
which are required in the equations. These are 
called the Besselian elements. 



588 Siddhanta Darpana 

(i) The elements d, x and y - Let the axes of 
x, y, z chosen as above meet the geocentric 
celestial sphere in X, Y, Z respectively. Let the 
right ascencion and declination of Z be (a,d) 

Then, as is evident from the figure, equatorial 
coordinates of X and Y are, (90 ° ■+ a, o) and 
(180 °+a, 90°-d). 

To find a and d, we note that x and y 
coordinates of the sun and the moon are same (for 
Z axis is parallel to SM) 

Let (d,<3) be the R.A. and declination of the 
sun and (di, d\) those of moon. If A is the sun's 
position on the celestial sphere, the values of 
Cos X A, Cos YA and Cos ZA can be easily 
written down. Thus, if (x, y, z) are the coordinates 
of the sun's centre S, and r is its distance from 
E, we have 

x = r cos XA =r cos <5 sin (a -a) 

y = r cos YA =r [Sin 6 cos d - cos d sin d. 
cos (ct-a)] 

z = r cos ZA = r [ sin <5 sin d + cos d 
cos d cos (a-a)] 

Similarly, coordinates (xi, yi, zi) of the moon 
are 

xi = ri cos 6 i sin ( a\ -a) 

yi = ri [Sin d\ cos d - cos diSin d cos («i-a] 

zi = ri [sin 6\ sin d + cos 6 cos d cos («i-a] 

where n is distance of moon's centre from E. 
Solving the equations obtained by putting x = xi 
and y = yi, we get a and d, the later being one 
of the Besselian elements. Substitution of these 



Solar Eclipse 589 

values in the expressions for x and y will give us 

x and y, the other two elements. 

Values of x and y are calculated at the interval 

of 10 minutes for tlie whole duration of the eclipse. 

Therefore, x' and y', the variations in x and y per 

minute can *ilso be easily determined. 

The elements x and y are obviously the 

coordinates of the centre of the shadow on the 
fundamental plane. 

(ii) The element fi - Let fi be the hour angle 
of Z from the meridian of Greenwich at the instant. 
The Greenwich sidereal time is g. Since the R.A. 
of Z is a, the value of p is G-a. After ft has been 
tabulated at intervals of 10 minutes, fi' (the 
variation of u per minute) can also be easily 
tabulated. 

(iii) The elements ft, f 2 - The semi vertical 
angles of the penumbral and umbral cones are 
denoted by ft and fe respectively. Now the radii 
of the sun and moon are R and b, and the distance 
between their centres is approximately r-ri; so ft 
and f2 are given by 

• c R + b • < , R - b 

sin f x = , sin f 2 = ~ — 7 

r - ri r - n 

(iv) The elements h, h - The radii of the 
circles in which the penumbral and umbral cones 
intersect the fundamental plane are denoted by h 
and h respectively. These also can fce fcmnd by 
simple geometry. 

Ii = b sec fi + zi tan ft 

and U = b sec k - zi tan h 



590 



Siddhanta Darpana 



where z a is the distance of the moon's centre 
from the fundamental plane and has been found 
in paragraph (i) above. 

In the Nautical Almahc, the quantities x, y, 
sin d, cos d, fi, li and h are tabulated at the 
intervals of 10 minutes for every solar eclipse. It 
is to be noted that these quantities relate to the whole 
of earth and not to any particular place on it. 

Circumstances of solar eclipse at a given place - 

Let p and O' be the geocentric distance and 
latitude of the place and x its longitude west of 
Greenwich . The hour angle of Z from the meridian 
of the place is fi-X since the hour angle of Z from 
Greenwich is j*. So if (§,17,?) are the coordinates 
of the place at any instant, we have 
£ = p cos <p' sin (a -A) 

77 = p [Sin (p f cos d - cos <p' sin d cos (a -A)] 
% = p [sin <p' sin d + cos <t>' cos d cos (w - A)] 
The values of ( £17, £ ) can be computed for 
any instant. Also, since p is the only variable in 
these expressions, formulae for ( §', r\\ % ) (the 
rates of changes of £, t], and, £ per minute,) cart 
be found by differentiation and the numerical 
values of ( ?', rf, ?' ) can be determined for the time 
of eclipse. 




fundamental 
plane 



Figure B - Elements of solar eclipse 



Solar Eclipse 591 

Consider now the sections of the penumbral 
and umbral cones by the plane z = £, i.e. the plane 
through the observer parallel to the fundamental 
plane. The sections will be circles; and if their radii 
are Li (for the penumbra) and L 2 (for the umbra), 
we have from the figure 

Li = li - 5 tan fi 

L 2 = k + £ tan f 2 

from which Li and L 2 can be determined. 

Consider now the beginning or the end of a 
partial eclipse at the given place. At these two 
instants, the point (§ ,rj£ ) must be at the distance 
Li from the axis of the shadow, which cuts the 
fundamental plane in the point (xi,yi,o) and 
therefore cuts the plane z = £ in the point x,y,£. 
The condition for this is 

(x- £ ) 2 + (y-vf = L i 2 - - - 0) 

Replacing x, y, £ and rj by x +x'lt and similar 
expressions, (1) becomes a quadratic in t. Solving 
it, we have the times for beginning and end of 
the partial eclipse. It we write L 2 for Li in (1), we 
can similarly determine the beginning and end of 
the total eclipse. In (1) it is sufficient to take the 
value of Li or (L 2 ) at an estimated time close to 
the time of occurrence of the eclipse, for Li and 
L 2 change very slowly. 

To determine the point on sun's disc where 
the eclipse begins - Figure 9 represents the 
penumbra section by the fundamental plane. C is 
centre and CX', CY' are parallel to the axes of x 
and y. Then the generator of the penumbra 
thorugh Y' touches the sun in the most northerly 
point because the earth's axis lies in the plane 



592 



Siddhanta Darpana 



x= o. Also, the generator through X' touches the 
disc in the most easterly point. Suppose that 
( £ rj £) lies on the generator through T. Then, if 
angle Y'CT = 

Li Sin 6 = (Xo+x't) - (go + g't ) 

Li cos 6 = (yo+y't) - too+^t) 

Substituting in it 
the values of t and the 
other quantities for the 
beginning or the end 
of the partial eclipse, 
we get the cor- 
responding value of 0, 
which is the position 
angle of the point 

where eclipse beginsFigure 9 - Starting point of solar eclipse 

or ends, because sun's 

disc is almost parallel to the fundamental plane. 

Verses 86-87 : Maximum and minimum values* 
of eclipse 




pala 



Maximum duration of candra grahana = 590 



Maximum duration of total lunar eclipse i.e. 
marda kala = 273 pala 

Maximum duration of solar eclipse = 632 pala 

Maximum duration of annular eclipse (valaya 
grasa) = 48 pala 

Maximum duration of total annular eclipse 
(marda kala)*= 23 pala 

Maximum increase in duration of a tithi = 405 
pala i.e. maximum value is (60+6/45) = 66/45 danda 



Solar Eclipse 593 

Maximum value of naksatra tithi = 67/45 danda 
Minimum value of naksatra tithi = 52/12 danda 

• ■ 

Maximum increase in yoga (beyond 60 danda) 
= 162 pala 

Maximum decrease in yoga = 664 pala 

Maximum gati phala of moon = + 7742 vikala 
or - 4927 vikala. 

Maximum gati phala of sun = + 123 vikala or 
- 117 vikala 

Maximum sphuta lambana = 5/12 ghati 

Notes (1) Maximum duration of lunar eclipse- 

The total duration of a lunar eclipse is given in 
hours 

V , 2 \ a [D 2 - P 2 (1 - -2^-1)1 1/2 
v p' 2 + m' 2 L p + m J 

where D is the distance between the centres 
of the moon and the shadow of first or last contact, 
P is the latitude of the moon at the time of 
opposition of the sun and the moon in longitude, 
p' is increase in P per hour and m' is the motion 
per hour in longitude of the moon, relative to the 
sun. 



This is clearly when D 2 = P 2 



1 - 



,2 
P 



i.e. when D ± P 



,2 
P 



,2 ,2 

p + m 



/ 



1 " ,2 "2 

V P +m , 



i.e. when P is numerically greater than D by 
Op' 2 / 2(p' 2 +m' 2 ) approximately. THis comes to about 
14" on the average. Thus even when P is greater 



594 Siddhdnta Darpana 

than D by upto 14", at conjunction, there can be 

eclipse. When P=D / duration of eclipse is not O, 

but 2Pp'/p' 2 +m' 2 , which is about 22 minutes. 

The duration is maximum, when latitude of 

the opposition P is o. It is equal to 2D/p' 2 +m' 2 
But D, m and p, are function of 1 and 1', 

mean anomalies of moon and sun respectively. 

Therefore, the maximum duration itself varies 

between limits. 

Let 1 and 1' be anomalies at sthula parva; time 

of fictitious conjunction or opposition or opposition 

between. 

True moon = mean moon + 315' sin 1 
True sun = mean sun + 127' sin 1' 
Equatorial horizontal parallax of moon 
II = 3447". 9 + 224".4 cos 1 
for sun, W = 8.8" + 2" cos 1 
Moon's semi diameter r = 939".6+61".l cos 1 
Sun's semi diameter r' = 961".2+16".l cos 1 
Radius of shadow S = 2545".4 + 228".9 cos 1 

- 16".2 cos 1' 

V m ' 2 + p ' 2 = 1875". 6 + 260".l cos 1 - 5".0 

cos r 

Now the distance between the centres of the 
moon and the shadow of first or last contact 

D = s+r - 3485". 0+290". cos 1 - 16.1" cos 1" 

2D 

and j |2 T 
m + p 

2 (3485".0 + 29(r\0 cosl - WW cosl') 

1875.6 + 260.1 cos 1 ~ 5"0 co^l' . 



hi 



t.">. 






Solar Eclipse 595 

This is a maximum when 1 = 1' = 180° and 
not when 1 = 1' = 0, as increase in denominator 
is more 

Thus maximum value is 

2(3438-290+16.1) hours = about 238 minutes 

1875.6-260.1+5 

This is correctly given as 590 pala = 5/2 X 238 
min. 

The lower limit occurs when 1 = 1'= and it 

is 

2 (3438 + 290-16.1) ^ " . L 

~~- — — — ~— — — r^ = about 212 minutes 
1875.6 4- 260.1-5 

If we do not neglect the function of 21, the 
maximum is about 237.4 minutes. 

Maximum duration of the total phase of a 
lunar eclipse is given by D = s-r. This also is 
maximum when sun andmoon are at opposition at 

Plhe nodes and when 1 = 1' = 180°. It is 

! 2 (1605.8 - 167.8 + 16.4) r _ g 

|l 1875.6 - 260.1 + 5 

I minutes 

I It is given in text as 273 pala = 0.4 X 273 = 

I 109.2 minutes 

' ■ : 

|||2) Maximum duration of solar eclipse - 

I The formula for duration of a solar eclipse in 

||gsneral on any place on earth (as opposed to the 
k " duration at any particular place) is the same as for 



&■■ 



/,;.■; 



fc- 



'i: 






duration of a lunar eclipse. Only difference is that 



fe D *= H-H' + r+r' 



!*,'.*.: 



596 Siddhanta Darpana 

and P is the latitude of moon at conjunction 
of the sun and moon in longitude. 

Here also the duration is not when p = D, 
but when P is numerically greater than D by about 
20". When D = ± P, the duration is about 33 
minutes 

The maximum duration of a general solar 
eclipse occurs when P = 0, i.e. when conjunction 
in longitude is at a node. It is given by 

2D 

-7—3 it hours 

v p + m 

2 (5339.9 + 285.5 cos 1 + 15.9 cos 1') 

= — ^ : hours 

1875.6 + 260.1 cos 1 - 5 cos 1 

This is maximum when 1 = 180° and l'=0 

2 (5339.9 - 285.5 + 15.9) u 

Thus it is — hours 

inus it is 1875 6 _ 2601 _ 5 

= 6 hours 18 minutes approx. 

= 378 X 5/2 pala = 945 pala 

Under this condition eclipse is annular. When 
2 1 term is not neglected maximum is about 6 hours 
16 minutes. 

The duration of a solar eclipse at a given place 

on the earth is given by — 3 jr~ corrected for 

& (P + m Y 2 

parallax which changes rapidly and varies from 

place to place. But the maximum duration occurs 

when the central eclipse is at apparent noon. At 

this time, apparent semi diameter of moon is r + 

aobut 16". Also at noon, the retardation in relative 

hourly motion of moon is maximum, causing 

increase in duration of eclipse. For an hour angle 



Solar Eclipse 597 

34° on both sides of noon, the average retardaiton 
is (850".3 + 55" .4 cos 1) per hour. Total duration 
is given by 

2 (r + 16" + r') 

— rr- 2 7 — - hourly retardation due to 

v p + m 



parallax 



2 (1917 + 61 cos 1 + 16 cos Y) 



(1875.6 + 260 cos 1 - 5 cos 1') - (850.3 + 55.4 cos 1) 

2 (1917 + 61 cos 1 + 16 cos Y) 

1025.3 + 204.7 cos 1 - 5 cos 1' 

When 1 = 180° and Y = 0, 

The maximum is about 4 hours 35 minutes = 
275X5/2 pala = 687 pala (Text gives 632 pala) 

This occurs when conjunction occurs at a 
Node, central eclipse falls at noon, 1 = 180° and 
i'= 

The maximum duration of the annular or total 
phase at a given place is also at apparant noon 
lor the same reason. As the period is very short, 
*ve take the motion per minute. The duration of 
to annular eclipse near noon is given by 

*-.". 2 (r - r - 16) 

(31".3 + 4".3 cos 1) - (15" + 1" cos 1) 



K' 



i?.. 






w 



= 2 (5.7 - 61 cos 1 + 16 cos Y) 

16.3 + 3.3 cos 1 
This is maximum when 1 = 180°, l'=0 
Thus it is about 13 minutes (34 pala approx) 



$gg Siddhdnta Darpana 

It is given 23 pala in the text. Minimum is 
clearly 0. 

The total phase is given by 

2 (r +16" - Q 

(31".3 + 4.3 cosl) - (15" + 1" cos 1) 
_ 2 (61.1 cos 1 - 16.1 cos T - 5.7) 

16.3 + 3.3 cos 1 
This is max when 1 = 0, 1' = 180° when it is 

2 X 71 ' 5 = about 7 minutes (= 17.5 pala) 
19.6 

This is not given in the text 

(3) Other limit : Other limits depend on the 
maximum and minimum values of speeds of moon 
and sun. First we change the maximum gati phala 
which is numdaparadhi x dainjka mean gat . 

Gati phala is + ve when manda paridhi is 
maximum at the end of odd quadrants. Hence 
maximum positive gati phala is more and negative 
gati phala is less. 

From this we get maximum and minimum 
gatis of sun and moon, by 

Max. gati = madhya gati + max. positive gati 
phala 

Minimum gati = madhya gati - maximum 
negative gati phala 

Minimum tithi = max (moon gat i - sun gati) 



Solar Eclipse 599 

12 ° 

Maximum tithi = — : " ~tt 

iV "*~" nun moon gati - max sun gati 

13" 20' 

Max yoga = mm (moon ^ + sun gati) 

1 3° 20' 

Minimum yoga = ; ~T~ 7[r 

j & max ( mon g atl + sun gatl ) 



13' 20' 
Maximum naksatra = 



min moon gati 



13° 20' 
Minimum naksatra = — 






maximum moon gati 

Mean values of moon and sun gati are 790/35 
and 59/8 Kala. There mandaparidhi at odd quad- 
rants is 12°6' and 31°30'. Mandaparidhi of sun at 
end of second quadrant is 12° 30' and at the end 
of 4th quadrant is 11° 54'. (Verses 95-96 of spastha 
dhikara, chapter 5). Moon's manda paridhi it is 
188,5 kala more at end of 1st quadrant and 
minimum is 188.5 kala less at end of 3rd quadrant. 

Verses 88-89 : Prayer and conclusion 

The god is worshipped in forms of Parvati, 
Surya, Siva and Ganesa and gives fortunes to 
devotees. He also changes moon into rahu (at the 
time of solar eclipse) and puts little knowing earthly 
creatures in confusion by covering sun like a flake 
of cloud. The same god may remove our troubles. 

Thus ends the ninth chapter describing solar 
eclipse in Siddhanta Darpana written for tally in 
observation and calculation and education of 
students by Sri Candrasekhara born in renowned 
royal family of Orissa. 



Chapter - 10 

PARILEKHA 

(Parilekha Varnana) 

Verse 1 : Scope - To show the direction of 
sparsa, madhya and moksa in surya and candra 
grahana clearly through diagrams, I explain the 
methods now. 

Verse 2-3 : Valana - An oblique ray of light 
bends in water but doesn't bend in vertical 
direction. Due to that reason, the size of sun and^ 
moon and sara of moon, remaining same, it looks- 
smaller in middle sky and bigger at horizon. Hence, 
earlier astronomers, changed the values of moon, 
sun earth's shadow in meridian depending on hara 
at that time. This is being explained now. 

Notes : Valana means bending. Light rays 
bend due to refraction, hence it is now called 
refraction effect. In appendix to Triprasnadhikara 
(chapter 7) this has been explained. If angle of 
incidence of light to a denser medium be i and 
angle of reflection be r then 

u = = a constant for the medium (1) 

sin r 

Hence the bending (i-r) increase with increase 
in angle of incidence. This angle is measured from 
perpendicular to the surface, hence in vertical 
direction there is no. bending. As we move towards 
horizon, the bending is more. Thus, at sunrise (or 



Parilekha 601 

setting time) its lower end is at horizon having 
90* natamsa and upper end has slightly less 
(90° -32') natamsa. Thus the lower end will be raised 
more compared to upper and it will be flattened 
and look more elliptical. 

The angle of bending or valana, R is 

R = K tan z' (2) 

where K = fi-\ and z' is apparent zenith 
distance 

Difference z-z' is proportional to K sec 2 z. dz 
= 32' (K sec 2 z) for sun. 

Hence apparent angular diameter is difference 
between apparent natamsa z and z' of upper and 
lower. 

Maximum refraction at horizon is about 35'. 
Its variation is very fast near horizon due to very 
high value of sec z near z = 90°. 

Natamsa Refraction 



K 


0° 


0" 


'<i ; ■ 


5° 


5" 




10° 


10" 


V; . 


15° 


16" 


a- ■ 


30° 


34" 


I. 


45° 


58" 


to-- 


60° 


1'41" 


mi 

Ml ■-■ ■ 

r 


80° 
85° 


5'19" 
9'51" 


Bns'-' 


88° 


18'16" 


K..* 


88°40' 


22'23" 




90° 


35' 



Thus apparent reduction in vertical angular 
diameter at horizon is about 5'. 



5Q2 Siddhanta Darpana 

If D is the average of two perpendicular 

angular diameters observed at vertical distance z, 

then the real diameter is 

D [1 + 1/2 K (1 + sec 2 z] (4) 

which is bigger than the observed. This means 

that observed diameter will decrease as Z increases 

and is minimum at horizon. 

Verses 4-5 : Value of angular measure for 
birhba. 

Unnata kala sanku of moon (for lunar eclipse) 
or sun (for solar eclipse) is calculated for middle 
time of eclipse. We add 10314 and divide the sum 
by trijya (3438) to get the hara or value of 1 angula 
in kala. On dividing the bimba of planets or 
shadow or sara of moon by this hara, we get their 
diameters is angula units. 

Alternatively, half day is multiplied by 3 and 
added to unnata, kala of moon (or sun) and divided 
by half day to get the same hara. Value of birhba 
and sara in angula units is obtained by dividing 
their values in kala by this hara. 

Alternatively, for rough calculation, bimba 
Kala is divided by 3 to get its value in angula. 

Notes : (1) Surya siddhanta assumes (Candra 
grahana verse 26) that the proportional angular 
diameter of a graha is 3 units at horizon, then it 
becomes 4 unit at vertical position i.e. increase in 
the ratio of 4/3. Bhaskaracarya and Lalla have 
assumed 2-1/2 : 3-1/2 increase i.e., in ratio of 7/5. 
Actual increase as we have seen after verse 3 is 
from (32'-5') to 32' in sun's bimba i.e. in ratio of 
32/27 = 1.2 approx. Thus the ratios 1.33 of surya 



Parilekha 603 

siddhanta and 1.4 of Bhaskara II are much higher 
than the true ratio. 

Another approximation is that the increase 
has been assumed proportional to the angular rise 
above horizon upto value of 90° rise to top 
position, where it is maximum. Angle of rise 0° = 
90°-z. Putting it in equation (4) above, apparent 
diameter is 

T 
D = ~ 

1 + - K (1 + cosec 2 6) 
2 v 

For = 0, lower term cosec °o = which is 
not correct approximation. However, the increase 
is in proportion to value of cosec and not 
proportional to as assumed. This is increase of 
average diameter. Vertical diameter will increase at 
double rate. 

(2) 1 angula = 3 Kala at horizon 
and = 4 Kala at vertical position 
Height is proportional to unnata sanku, as 
assumed. 

For height of R (Trijya = 3438') increase is 1 



kala, 



U 
increase is — Kala 

Thus 1 angula = 3 + — Kala 

_3R±U Kal5 = 3 X 3438 + U^ 
R ^^ 3438 

U + 10314 _, _ 

= — Kala 

3438 



604 Siddhanta Darpana 

This is the first formula 

Roughly half day is of 15 ghati when sun 
reaches at top. Actually it is still slightly away from 
zenith but that distance is ignored. Unnata kala is 
in proportion to half day taken as 90° or 15 ghati. 

Unnata kala LI 

Hence ' half day = ¥ 

U Unnata Kala 

or, langula = 3 + - = 3+ ^ ^ y 

3 x half day + unnata kala 

half day 

This is alternative formula 

If we totally ignore the variation due to 
refraction, except for horizon position, diameter is 
almost same, and 1 angula = 3 kala is uniformly 
assumed. 

bimba in Kala . . . 

Thus =r~rr. — ~ r~ = bimba in angula 

Kala in 1 angula 

Verses 6-14 : Diagram for direction of eclipse 

On a ground, plane like water level, a circle 
of 57/18 angula semi-diameter is drawn with a 
compass. This is known as khagola vrtta having 
two valanas. 

From this centre only, another circle with 
radius of sum of semi diameters is also drawn 
which is called samasa vrtta. 

From same centre a third circle is drawn with 
radius equal to the grahya bimba (which is eclipsed) 

Now according to method explained in 
Triprasnadhikara north south line and east west 
lines are drawn in khagola vrtta. In lunar eclipse; 



Parilekha 6°5 

sparsa is from east and moksa is in west direction. 
But in solar eclipse sparsa (beginning) is from west 
and moksa is in east direction. 

In khagola vrtta we mark a point at a distance 
from east point for lunar eclipse equal to jya of 
sphuta valana and in same direction as valana. A 
line from centre to that point is drawn. Similarly, 
at a distance from west point equal to and in 
direction of moksa time valana, another point is 
chosen and a line from centre is drawn. In solar 
eclipse, the order of valana lines is reverse i.e. 
sparsa in west and moksa in east direction. These 
lines are called valanagra rekha. Valanagra rekha 
cuts samasa vrtta on valana points. From these 
points, we mark the distance equal to sphuta sara 
jya of moon at the time of sparsa or moksa. These 
are called saragra vindu (in east for sparsa and 
nimHana and west for unmllana and moksa in lunar 
eclipse, opposite direction in solar eclipse). 

The line from centre to saragra point cuts 
grahya and moksa. Here sara and valana are given 
according to their current values. 

Sara is in north south direction, some times 
in angle direction like agni kona (north east). 

Notes : (1) Radius of khagola vrtta is 57/18 
angula because 57° 18' = 3438' = length of radius. 
Hence 1/60 angula on radius or circumference is 
equal to 1 minute or kala. The method is same as 
in surya siddhanta, but there the radius is 49 
angula where 1 angula was 70'. 

Radius of samasa vrtta or grahya vrtta will be 
calculated according to value of angula in kala 



606 



Siddhanta Darpana 



calculated in verses 4-5. Roughly 1 angula = 3 kala. 
Similarly length of sara also is calculated in angula. 

However, valana is measured on khagola vrtta 
where 1 pratyangula (1/60 angula) is equal to 1 
kala or 1 angula = 1°. With this unit we measure 
the lengths. 

(2) Method of drawing is best explained by 
actual diagram. 




Figure 1 - Diagram for sparsa and moksa in eclipse 

ENWS - Khagola circle, 1 angula = 1°, 57/18 
radius, E'N'W'S' - samasa vrtta, E,E' east points, 
N, N', North points; W, W west points S,S' South 
points, AB is grahya birhba 

EVi = Valana jya for sparsa in solar, moksa 
in lunar eclipse. WV2 = Valana jya for moksa in 
solar eclipse, sparsa in innar eclipse Vi'S = Current 
sara of moon 

Vi'S' current sara 

Vi', Vz are lines on samasa vrtta cut by OVi, 
OV 2 . OS; OS' cut grahya on A,B which are points 
of contact. 



Parilekha 607 

Verse 15-30 : Further details for periods 
within sparsa and moksa. Now, I describe the 
details of eclipse between the end points of sparsa 
and moksa. 

In lunar eclipse, when moon is near rahu or 
ketu, spasta valana in khavrtta is given in own 
direction from east or west point in north or south 
direction. From these valana end points, we give 
two points at distance of 5 angula, in north 
direction from east valana, and south direction from 
west valana point. We draw a line through these 
points which also passes through centre of the 
circle. 

In solar eclipse, we mark a point from eastern 
valana point at a distance equal to lagna kranti in 
the direction of kranti. This point is joined with 
centre and extended to make it diameter. Sara of 
moon is put in perpendicular direction on its line 
according to direction of the sara. (Sara will be at.\ 
central point for middle position of the eclipse or 
any other point according to time of eclipse). From 
end point of sara a circle is drawn with radius of 
grahaka bimba (eclipser) (This circle is drawn in 
lunar eclipse on 5° difference line). 

The portion cut by grahaka bimba will be the 
extent of eclipse visible to people. 

Sara of sparsa, madhya and moksa periods 
are put at their positions. From the three end 
points of sara, we draw three circles with radius 
egiial to 1/3 of the distance between sparsa and 
From intersection of adjacent triangles two 

like figures are formed. The head tail lines of 
£hese fish figures join at a point which is centre 



608 Siddhdnta Darpana 

of circle passing through these points. With this 
centre an arc is drawn through sara ends of sparsa, 
madhya and moksa which is the grahaka marga 
(path of the eclipsing planet or shadow). 

From centre of this grahaka marga, we draw 
a line in the direction of sparsa (eastern direction 
in lunar eclipse and west in solar eclipse), at the 
distance of grahaka diameter from sparsa point, 
there will be nimflana point on the grahya circle. 
Similarly, unmilana point on grahya circle will be 
on the moksa side of the grahaka marga. 

To find the amount of grasa at desired time 
we assume two parts of grahaka marga - from mid 
point to sparsa, it will be sparsa khanda and the 
other side will be moksa khanda. Their length is 
measured in angulas. The angula measure is 
multiplied by required time (after sparsa or before 
moksa) and divided by its sthiti ardha time. We 
give. a point at a distance equal to angula measure 
of required time from sparsa or moksa point. From 
that point, we draw a circle with radius of grahaka 
circle. The portion cut by this circle in the grahya 
birhba will be the required amount of grasa at 
desired time. 

Sum of semi diameters of grahya and grahaka 
is substracted from the required grasa in angula. 
A pointer equal to remaining length in angula is 
taken. With this, we find two points on grahaka 
marga at distance of grasa from centre of grahya 
circle. One point is in sparsa khanda and the other 
in moksa khanda. From these points we draw circle 
with radius of grahaka birhba. The portion covered 
by this circle will be the portion eclipsed. 



Parilekha 609 

At the distance of difference of semi-diameters 
of grahya and grahaka from centre of grahya birhba, 
we get two points on grahaka marga - one on 
moksa khanda and the other on sparsa khanda. 
These are the points of nimilana (on sparsa khanda) 
and unmilana. 

Notes : 




Figure 2 - Diagram for sparsa and moksa in eclipse 

ENWS are direction points on khagola circle, 
1° = 1 angula; Radius = 3438' = 57° 18' = 57/18 
ahgula. 

E'N'W'S' - Samasa circle direction points, 

radius equal to sum of semi diameters of grahya 

and grahaka. For bimba and sara length, 1 angula 

unnata sanku kala 

= 3 + --— 

Tnjya 

Vi, V2 are valana points. EVi and WV2 are 
^qual to magnitude of direction of valana jya. 

V1L1 = V2L2 = 5* i.e. 5 angula on Khagola 
circle which is equal to inclination of moon's orbit 
with ecliptic. Thus Li L 2 is path of moon for lunar 
eclipse. 



g20 Siddhdnta Darpana 

For solar eclipse V1L1 = V 2 L 2 = kranti of lagna. 
On its intersection with samasa circle and at 
centre, sara lengths at sparsa, madhya and moksa 
points are drawn, perpendicular to it. It will be 
least at the centre and in direction of sara at all 
places. Their ends are Si, S 2 , S 3 : The three circles 
through these points from two fish figures which 
intersect at point C. From C as centre with radius 
CSi = CS 2 = CS 3 we draw a circle. Si S 2 S3 arc is 
the grahaka marga on which eclipser planet or 
shadow moves. 

For lunar eclipse S 2 S3 is sparsa khanda and 
Si S 2 is moksa khanda. For solar eclipse Si S 2 is 
sparsa khanda and S 2 S3 is moksa khanda. 

Nimflana point P for lunar eclipse (or un- 
milana point for solar eclipse) is on grahaka marga 
such that S3 P = diameter of grahaka birhba. 

Length on grahaka marga is proportional to 
time. 

Hence for any point P 

Length Desired time 

S 2 S 3 " Sthiti ardha 

This formula is used to calculate grasa at 
desired time. 

Verse 30-35 : Another method of diagram - 
At saragra point on one side of valanagara 
rekha (sara is madhya sara), another line parallel 
to valana rekha is drawn. From its end points on 
khagola circle, a ppint is given towards north for 
lunar eclipse (south for solar eclipse) at a distance 
of 1/60 of Jya of local aksarhsa. 



Parilekha 611 

From these two points and the point of 
madhya sara point (i.e. mid point of parallel line 
to valana rekha) we draw a circle as explained in 
above verse. 

Portion of this circle within samasa circle will 
be grahaka marga. On this path, we can find 
nimilana and unmllana points from centre of grahya 
circle at distance of difference of semi-diameters of 
grahya-grahaka, . as before. In this diagram sparsa 
and nimilana of solar eclipse can be seen in west 
direction and, for lunar eclipse in opposite direction 
very easily. This method doesn't need sara or 
valana time at time of sparsa, etc. 

But,, for diagram of solar eclipse, 1/3 of sara 
of moon (i.e. angula value) is kept at two places. 
At one place it is multiplied by sun sanku of that 
time and divided by 4400. Quotient is added at 
first place. 

On a single board both solar and lunar 
eclipses can be shown. Only difference will be that 
the direction of sparsa, moksa etc will be opposite 
for the two types of eclipses. 

Note : This is almost same procedure. In stead 
of marking sara at sparsa, moksa and mid points, 
we mark the middle sara only. In stead of other 
sara, we mark the kranti of lagna on khagola at 
distance from middle sara. Reason is that the 
diurnal circle of moon will be parallel to ecliptic 
and at same angular distance from lagna point of 
ecliptic as on middle point of eclipse. 

For solar eclipse sara is corrected for parallax. 
The correction is slightly less, which appears to 



g22 Siddhanta Darpana 

compensate effective increase of tamo-mana of 
moon as explained in chapter 9 verses 43-45. 

Figure 1 and 2 show, that both the diagrams 
for solar and lunar eclipse can be combined, which 
has been prescribed here. 

Verses 37-38 : Prayer and conclusion 

I pray to lord Jagannatha, who smiles with 
beautiful lips, beauty of whose round eyes defeats 
the beauty of morning sun of spring time and full 
moon of winter night, who gives freedom from 
fear to people flocking to Nilacala from different 
regions, and whose sight can emancipate the world. 

Thus ends the tenth chapter on diagrams in 
siddhanta darpana wirtten for calculation according 
to observation and instruction to students by Sri 
Candrasekhara, born in famous royal family of 

Orissa. 



'■■«'*' 



Chapter - 11 

CONJUNCTION OF PLANETS 

GRAHA YUTI VARNANA 

(Conjunction of planets) 

Verses 1-2-Scope - While the planets are 
moving in their own orbits, their position is seen 
same from earth. This is called graha yuti (con- 
junction of planets). Graha yuti and its good or 
bad results are described in this chapter. 

According to Surya siddhanta, when tara 
graha (mangala etc.) are seen joint, then their 
(apparent) . coming together is called graha yuti or 
yuddha. When any tara graha comes together with 
moon, it is called samagama. When tara graha is 
with sun, it is not visible due to bright rays of 
sun, and it is called 'asta mita' (heliacal setting of 
planets). 

Notes (1) Planets do not really come together. 
They are in their own orbit which are far from 
each other. But due to parallax, they are seen 
together, as in solar eclipse, sun and moon are 
seen in same direction. However, the parallax is 
same for all positions from earth due to large 
distances of star like planets (tara graha). Compared 
% eclipse of sun, the diameters of tara graha are 
^uch smaller and their orbits are farther and 

r, hence their conjunctions are rare. However, 







624 Siddhanta Darpana 

their number is more causing different combina- 
tions of yuti and their sara also is small compared 
to moon's orbit, so we are able to see the yuti 
some times. 

(2) Moon is considered the king of stars and 
the naksatras as its wives. It lives with one naksatra 
each day like a husband and wife - 'naksate' means 
lives together. Thus conjunction of moon with any 
naksatra or tara graha is called samagama or happy 
union. Conjunction between tara graha is called 
'yuddha, as it is not considered friendly. In this 
'yuddha ' or war, the planet which is behind is 
like a chaser and takes away half the strength of 
the other planet which is considered defeated. This 
strength is considered in astrology for considering 
their power in giving good or bad results. The 
reduction or increase of strength is according to 
their mutual covering and depends on their angular 
diameters. At present, we follow the method of 
Sripati for calculating the reduction or increase in 
strength due to planetary war. 

Due to nearness with sun, the planets are 
invisible and called set due to sun. This has already 
been mentioned in chapter 6 and will be discussed 
in an independent chapter on it. 

(3) Varahamihira in his Brhat samhita, ex- 
plained in detail the various results of graha yuti. 
According to the degree of their seeming approach- 
ment, there are four kinds of wars (among planets) 
as stated by Parasara and other sages - Bheda 
(occulation or cleaving), Ullekha (grazing), 
Arhsu mardana (clashing of rays) and Apasavya 
(passing south ward). 



Ifrrijunction of Planets 615 

Yeises 3-5 : Principles of computation 

We find rasi, amsa and kala of two planets 
in conjunction. When they are equal in ecliptic 
/kadamba prota vrtta), their values on equator are 
found (dhruva prota vrtta). From this, their sara 
and lambana are found. Then bimba (angular 
diameter) is calculated. 

In surya siddhanta - When faster of the two 
planets has greater longitude (i.e. it is towards 
east), then conjunction has already occurred. If it 
i*fe less (i.e. in west), then the conjunction is yet 
% occur. If both are vakri (retrograde) then reverse 
*oll happen, i.e. planet in east indicates, conjunc- 
tion is to occur, in west means conjunction has 
already passed. If one body alone is retrograde 
laid its longitude is greater (in east), then the 
conjunction is to come, if less, it has passed. 

Notes : (1) Conjunction is calculated first in 
longitude measured along ecliptic, when their 
positions are same. However, their difference in 
perpendicular direction (sara) and apparent devia- 
tion due to observing from earth will depend on 
position with respect to equator. Size of the bimba 
of planets will decide, at what distance they will 
meet. 

(2) Finding conjunction, time, whether gone 
or yet to come is very easy to find, from diagram. 



ft 







Figure 1 - Conjunction of planets 



616 Siddhanta Darpana 

In figure 1, M is position on ecliptic, which 
is mesa 0° from which position of planet is 
measured. When arrow direction indicates rotation 
in east direction, the rasi arhsa etc (longitude) of 
two points A and B are MA and MB. When 
longitude of B is more, it is east from A as seen 
from figure.. 

When B is faster, it will move further east 
from A, and at some earlier time it was with A 
i.e. in conjunction. If A or western planet is faster 
it will meet B in time needed to cover AB with 
relative speed. If B is retrograde in east position 
A and B, both approach each other with their 
speeds, hence it will approach with speed equal 
to sum of speeds. When both are moving in 
western direction, obviously the reverse of direct 
motion will happen. 

Verses 6-9 : Finding the time and place of 
conjunction - At required time we find the 
bhogamsa (longitude) of the two graha and convert 
their difference into kala. This is separately 
multiplied by daily speeds of graha in kala. Each 
product is divided by difference of speeds if both 
have direct or both retrograde motion. But if one 
graha is margi and the other, vakri, the products 
are divided by sum of speeds in kala. If both 
planets have already joined and both are margi, 
then each quotient is deducted from the bhogamsa 
of its planet by whose speed it had been multiplied. 
If conjunction is yet to happen, then the quotients 
are added. If both are retrograde (vakri), reverse 
is done. If one is vakri and the other margi, then 
addition and substractipn are done as per rules 
explained earlier. By this, we get the bhogamsa of 



Conjunction of Planets 617 

Kranti vrtta (position on ecliptic) where conjunction 
w has happened. If the kala of planets doesn't become 
j equal in a single operation, this process is repeated 
I again. 

Notes : In figure 1, 

longitude of A is MA, B is MB 

Differnece in longitudes is MB-MA = AB 

Speed of A is a and B is b kala per day 

Difference in speeds is a-b 

if a > b, then A will catch up with B in time 
AB / (a-b) 

if a, < b then B has gone ahead this difference 
I AB in time AB / (b-a) 

Thus in first case the longitude of conjunction 

AB 

for A will increase X a, increase in B will 

a-b 

be x b. This increase in A will be (a-b) 

a-b a-b 

i. " L 

more i.e. AB more and they will catch up. 

If a < b, then the conjunction time is earlier 
I and longitude of A and B will be reduced by 
| distances travelled by them. 

For retrograde planets obviously situation will 
I be reversed. If B is faster, it will catch up distance 
g . BA in time t = BA / (b-a) in which the longitude 
of B and A will be reduced by t b and t a. 

Suppose A is retrograde and B is forward 
motion. There relative speed is at a+b and their 
distance is increasing. Then they are together at 
time AB / (a+b) = t before the present time. In 
this time longitude of A was more by ta because 









5ig Siddhdnta Darpana 

it is retrograde and B was tb less, in earlier time 
of conjunction. 

If A is direct and B is retrograde, then the 
plenets are approaching each other with velocity 
a+b and they will cover the distance AB in time 
AB / (a+b) = t when they will be together. After 
that time position of A will be ta more and of B 
will be tb less because it is moving in reverse 
direction. 

(2) We are assuming uniform motion of 
planets in the interval AB. Within this the speeds 
will change, forward motion may become 
retrograde and vice verse. Thus after getting the 
conjunction time approximately on basis of present 
speeds, we again calculate the position difference 
at this approximate conjunction time. Then we 
calculate more accurately as to when conjunction 
had occurred or would occur. 

Verses 10-11 : Sara of planets 

(From Surya siddhanta - Spastadhikara verse 
56-57). 

In pata of mangala, sani and guru, correction 
for second sighraphala is made in same manner, 
in which it is done for the planet (i.e. positive 
result is added and negative substracted). This will 
give the true postions of pata of these three planets. 
But in pata of budha and sukra, correction is made 
with second mandaphala (used in third step of 
correction) in reverse manner - i.e. positive result 
is substracted and negative added. By this, true 
pata of budha and sukra will be known. 



Conjunction of Planets 6W 

From true postions of mangala, sani and guru, 
true positions of their pata are deducted to get 
viksepa kendra. Viksepa kendra of budha and 
gukra are found by substracting their true pata 
from their slghrocca positions. 

Jya of viksepa kendra is multiplied by madhya 
viksepa and divided by fourth slghra karna to get 
the sphuta sara. 

Notes : (1) Mean inclinations (viksepa) of 

planetary orbits - This has been explained by 

Bhaskaracarya II. in his chapter on 

grahacchayadhikara (siddhanta siromani). Reasons 

| of the method have also been explained. 

The values of madhya viksepa are given in 
chapter 5 - spastadhikara verses 28-33, reproduced 

here 

Planet Siddhanta Darpana Modern value 

value 

Candra 5'9' 5'8'42" 

Mangala 1*51' 1"51'<T 

Budha 2°44' 7°0'14" 

Guru ris- ri8'2r 

Sukra V 28' 3°23'39' 

Sani 2"29' 2 e 29'25" 

The values of superior planets are almost same 
as modern values. Bhaskara says that these values 
are for that time when slghra anomaly is equal to 
90° + 1/2 R sin _1 a, where a is R sine of the 
maximum slghra phala. This is quite correct, 



// 



iff 



620 



Siddkanta Darpana 





E "A' 

Figure 4 Figure 5 

' because when the 
sighra anomaly has this value, the true planet is 
at point of intersection (P) of the deferent and 
eccentric circle. Then the planet is equidistant from 
Ei and E2 (figure 2) For superior planets, Ei is 
taken as earth's centre and E2 is sun, the mean 
latitude of the planet observed will be same, 
whether observed from earth or sun. Hence, 
maximum latitudes of the superior planets are same 
for geocentric and heliocentric observations. These 
are the mean values. 

For inferior planets, mean planet in this case 
is taken to be sun, the linear values of the latitude 
observed from E and S, the centres of Earth and 
sun will be in ratio SP/ES (figure 3) For mercury 



Conjunction of Planets $21 

this ratio is 4/10 and for venus it is 7/10. Hence 

420 X 4 
the modern values are reduced to and 

10 

204 x 7 . 

- — — i.e. 168 and 142 which are approximately 

equal with the values given in siddhanta. 

(2) Pata is calculated for orbit round sun and 
converted to geocentric position - 

Figure 4 shows an inferior planet indicated 
by P and Figure 5 an superior planet J. Position 
of earth and sun are E and S., position of mesa 
0° from sun and earth are A and A'. Position of 
node from sun and earth is N and N'. 

True position of inferior planet is P and 
superior planet is J. U is the mandocca position 
(i.e. sun) for P. 

Pata of inferior planet - 

Convex angle ASN is heliocentric longitude 
of node measured negatively, as node has a 
negative motion on ecliptic. Rule says that 
heliocentric sighra anomaly is added to this which 




Convex angle ASN + ZUSP = 360° + ZNSP 
- Z.ASU = ZNSP~Z.ASU 

Now longitude of planets is added here i.e. 
fUASU (= ^A'ES) 

p Result is Z.NSP. 

pi,-. ;> . 

<*.* , . RSinNSP x B 
bara as seen from sun is — — 

R 

P is maximum sara (latitude). 



I 




522 Siddhanta Darpana 

As seen from earth this is to be reduced in 
ratio R/K where K is distance from earth i.e. 
sighrakarna. 

Thus sara seen from earth 

RSi " NSP X g which is the formula. 
K 

Pata of superior planet - True geocentric 
longitude of J is ^A'EJ = Z.ASJ' 

Substracting slghraphala EJS = JEJ' from this 
we get Z.ASJ = heliocentric longitude. 

Then retrograde longitude of N i.e. Z.ASN 
is added. 

We get Z.ASN + L ASJ = ^NSJ . From this 
heliocentric sara (latitude) is first calculated as in 
above case by multipliying with (MR and then 
geocentric valkue is obtained by R/K. 

^NSP or ZNSJ has been called viksepa 
kendra i.e. heliocentric distance between pata and 
planet in both cases. 

Verses 12-26 : Further correction for sara - 

The above sara has been written according to 
old siddhanta which is inaccurate according to 
author. Now accurate sara of mangala etc as 
actually seen in explained. 

Sun and moon are to be corrected for parallax, 
when away from midday-sun (i.e. zenith), due to 
difference of observation from earth's centre and 
surface. Similarly, correction in sara is to make it 
sphuta (from heliocentric to geocentric position). 

Mean positions of mangala, guru and sani are 
substracted from their sphuta mandocca to get the 
manda kendra. Jya of manda kendra is mandaphala 



Conjunction of Planets 623 

approximately. By adding or substracting this from 
mean position we get manda sphuta graha. 

Sighrocca of budha and sukra is substracted 
from their mandocca sphuta. For budha, its 
gighrocca is corrected by its parocca kandraphala. 
Result is sighra kendra for viksepa purpose. 

For viksepa kendra of other three planets, 
manda spasta graha is substracted from its pata. 

These are sara kendra of all 5 planets. From 
its bhuja jya, sara is found by multiplying with 
parama sara and dividing with trijya - heliocentric 
value. Sara is in north or south direction as 
explained in case of moon. 

Difference of third mandakarna and trijya is 
multiplied by difference of fourth sighra karna and 
trijya and divided by trijya. We get ksepa 
karnantara. 

Sara Karna - (1) When fourth sighra karna is 
more than trijya - (a) when third manda karna also 
is bigger than trijya - Karnantara is substracted 
from trijya (b) when third manda karna is less 
than trijya - ksepa karnantara is added to trijya. 

(2) When fourth sighra karna is less than 
trijya- (a) manda karna is more - then karnantara 
is added to trijya (b) when mandakarna is less - 
then karnantara is substracted from trijya. 

For budha and sukra, karnantara is added or 
substracted from mandakarna instead of trijya. 

Thus we get sara karna of all the five planets 
for all situations. 



624 Siddhdnta Darpana 

Sphuta sara : As in previous method, pata is 
substracted from graha. Jya of this viksepa kendra 
is multiplied by madhyama sara and divided by 
sara karna. Quotient is multiplied by trijya and 
divided by fourth sighra karna to obtain sphuta 
sara of planets in kali. Its difference with sthula 
(rough) sara also can be used. 

Notes : (1) First we calculate the heliocentric 
position by mandasphuta graha as explained in 
spastadhikara. 

(2) Sara karna is real distance of planet from 
sun due to sara in its sighra gati. Difference of 
manda karna and trijya is proportional change of 
distance due to mandaphala. It is multiplied by 
proportional change due to sighra phala by 
multiplying with (sighra karna trijya) and dividing 
by trijya. 

When mandakarna or sighra karna is greater 
than trijya, sara karna i.e. true position of planet 
with sara, is less because sara will look smaller 
from larger distance. Hence sara karnantara is 
substracted from trijya, average distance. 

For budha and sukra average distance is their 
manda karna i.e. distance of sun from earth. 

« 

(3) Madhyama sara is value of sara seen from 
sun, it is multiplied by sara karna to get its true 
value as seen from sun. For proportionate reduction 
for geocentric value; it is multiplied by R/K. as 
explained in notes after pervious verse. 

Verses 27-31 : Ayana drk-karma : 

Sayana graha is added with 3 rasi (90 ° ) - 
which is satribha sayana sphuta graha. Its kranti 



Conjunction of Planets $ 2 5 

jya is multiplied by sphuta sara and divided by 
dyujya of satribha sayana graha. The result will be 
in lipta etc. and is called ayana drkkarma kala. 

When ayana and sara of graha are in different 
direction, ayana is added to graha; and substracted 
if they are in same direction. Then graha postition 
or equator will be found, i.e. kadambaprota graha 
will become dhruva prota. This is called ayana 
drkkarma. 

After doing ayana drkkarma, again the dif- 
ference of planets involved in war (conjunction) is 
found. As before; the time is calculated when their 
rasi, kala etc. are equal. This will give lapsed or 
remaining days of conjunction. At the time of this 
conjunction, the planets are equal upto kalas. Then 
again sara is found; ayana drkkarma for new 
position will be done. By repeating the process, 
we get accurate time of equatorial conjunction when 
kala of the two planets are equal. 

Notes : In figure 6, EMQ and CMD are nadl 
?mandala and kranti mandala respectively. P is 
dhruva, K kadamba and G the planet or 
grahabimha. PGA is 
dhruvaprota and KGB. 
kadambaprota. Then B is 
sphutagraha or position of the 
planet on kranti mandala. A 
;ls called krta ayana drkkarmaE 
jgraha - i.e. point on ecliptic 
^corresponding to equator 
pposition. MA may be called C 

jpolar longitude of the planet 

U modern terms. GB is Fjgure 6 . ^^ valana 







$26 Siddhdnta Darpana 

viksepa of G, which is almost equal to GA. 

From GAB considered plane triangle 

AB = M GXGA 

JyaB 

But, in GKP, LGKP = 90° + say ana graha = 
satribha graha, PK is measure of obliquity of ecliptic 
or parama kranti. 

x - ^ JyaGKP x JyaPK 
* a G = fi^PG 

Jya (satribha graha) x Parama Kranti Jya 

Dyujya 
Krantijya (satribha graha) x Trijya 

Dyujya 

Jya B = Trijya, as B = 90° in (1) 

Hence from (1) and (2) 

GA X Kranti jya of satribha graha 

Dyujya 

AB = ayana drkkarma, i.e. shift in position 
of planet on ecliptic due to inclination of axis and 
sara. 

Verse 31-37 : Aksa drkkarma - 

Square of ayana drkkarma in kala and square 
of sara are added. Square root of sum is the suksma 
sara. When suksma sara and kranti are in same 
direction they are added; otherwise difference is 
taken for sphuta kranti of the planet. This will be 
distance from planet to the equator on polar circle. 
Sun is always on kranti vrtta so its madhya kranti 
and sphuta kranti are same. 

By the method explained in Triprasnadhikara, 
for both the planets (in conjunction), from sphuta 
kranti, we find their cara, dinardha nata and 



Conjunction of Planets 627 

unnata kala. Nata and unata kala separately 
multiplied by 5400 and divided by their half day 
give jya of nata and unnata kala respectively. 
Difference of cara asu of graha for madhyama and 
sphuta kranti is taken as kala and multiplied by 
jttta jya and divided by trijya. The result in kala 
is substracted from graha in forenoon (east half of 
sky) and added to graha in west half, if sara is 
north. For south sara, reverse process will be done. 
Then the graha will be corrected with aksa 
drkkarma. 

After that, difference of both graha is found 
and the time since conjunction or remaining till 
that is found. For conjunction time; again aksa 
drkkarma is done. After repeated procedure, both 
graha will be in same samaprota vrtta. Then, their 
north south difference in found on that circle. 

Notes : (1) Sphuta sara : Sara (or madhyama 
sara) is GB in figure 6 which is distance of the 
planet from ecliptic along the circle through 
kadamba K. Along this circle the distance of planet 
from equator is GB'. But distance from equator is 
calculated along . great circle through dhruva P. 
Hence the total kranti i.e. distance from equator 
is 

GA' = GA + AA'. We take as spasta graha, 
i not real planet G but its projection B on ecliptic. 

Hence, kranti of B is the real kranti. 
?' ; First we have to calculate GA, which is given 
by GA = VqB 2 + AB 2 as ^GBA is 90° and 
AGBA is small and considered a plane triangle. 
f AA' is almost equal to BB' which is kranti of 

the sphuta planet i..e madhya kranti. 



628 



Siddhdnta Darpana 



Calculation of GA is really not necessary by 
the above formula, as we have already assumed 
GA = GB in derivation. 

(2) Bhaskaracarya has explained the drkkarma 
with difference in rising time on horizon due to 
sara of the planet. When the ecliptic position of 
the planet is rising on horizon, then due to sara, 
the real planet is above the horizon for north sara 
(down for south sara) and rises earlier (or later for 
south sara). The difference in rising time is known 
by drkkarma. One component of drkkarma 
depends upon ayana valana (i.e. inclination be- 
tween equator and ecliptic) and the other com- 
ponent depends on aksavalana (i.e. local aksamsa 
- inclination of local horizon or vertical with horizon 
or vertical of equator). These components are called 
ayana drkkarma and aksa drkarma. 




Figure 7 - Akla drkkarma 

In figure 7, NZSZ' is yamyottara vrtta of a 
place and NES is diameter of horizon in its plane. 
QQ' and AA' are diameters of equator and ahoratra 
vrtta. PP' is diameter of unmandala. EM is kranti 
jyi; AM is dyujya = R cos0 , where <p is aksamsa. 



Conjunction of Planets $29 

Due to kranti, the planet rises earlier at 
position N, MN = kujya = R sin d tan0 where 6 
is kranti. Its value on equator is ET where 1 kala 
= 1 asu in time. ET = carajya = R tand tan <p 

Due to sara, the planet at M on ecliptic is 
seen at K' in direction K, which is kadamba or 
pole of the ecliptic. (Sara is shown north, when K 
is north from P). LYMP = v = ayana valana. Thus 
due to sara, the longitude of planet is shifted by 
K'M' on diurnal circle, K'M' = s sin v where s is 
the sara. This is equivalent to shift of 

s Sin v I cos <p on equator, which is ayana 
drkkarma in asu. If we put R cos0 = Dyujya and 
R sin v = satribha kranti jya (approx), we get the 
formula for ayana drkkarma given earlier. 

Another component of sara MK' is MM' 
= s cos v, which is the sara in perpendicular 
direction to equator. Hence, sphuta sara is EM' = 
EM + MM' = R sin 6 + s cos v. Thus effectively 
the diurnal circle of ecliptic planet of M will be 
shifted to LM'L' parallel circle to equator passing 
through M'. Then the planet, will rise at position 
R' (corresponding to R on diurnal circle and T' on 
equator). Thus the rising time will be earlier by 
TT\ J 

XT' = ET' - ET = difference of carajya. This 
is the simplest and most accurate formula given in 
any siddhanta text. 

(3) Difference in carajya is difference at 
norizon, corresponding to half day length (dyujya 
<* radius of equator). At other times it is 
Proportional to natajya i.e. distance from 
yamyottara position A or L of the planet. Thus 

Aksa valana at ista kala _ Aksa drkkarma at rising 
Jya of nata kala ~ Half day 






$30 Siddhdnta Darpana 

Formula for aksa drkkarma at any other time 
has not been given by any other author. It is seen 
that aksa drkkarma is deducted from planet in east 
sky as it rises earlier for north sara. Since it will 
set later in west, half proportionate addition will 
be done. 

Verses 38-40 : Biihba of planets 

Five tara grahas like mangala have five types 
of birhba - madhya vrtta biihba, madhya bhasvara 
bimba, sphuta vrtta bimba, sphuta bhasvara birhba 
and drktulya birhba. 

Bimba of sun is very bright. Planets like moon 
take light from that and reflect it like water surface. 

Tara graha also have horns, due to the angle 

between direction of the graha and sun. But due 

to their distance from sun being large compared 

to moon, their horns are not seen. They are seen 

. as point only. 

Notes : (1) Birhba of tara grahas have been 
discussed in detail in birhbadhikara of siddhanta 
tattwa viveka, but this terminology has not been 
used any where. They are lighted by reflected light 
of sun, and bimba of sukra and budha are seen 
less than half when they are between earth and 
sun, due to their dark phase like moon. It has 
also discussed hole is sun due to sukra (like eclipse 
by moon). Due to smallness of taragrahas (small 
angular diameters) they are seen only as a point 
and their horns are not seen due to dark phase 
like moon. 

(2) From the context, the classification of 
bimba depends on their distance from earth, due 
to which they look small or big and due to phase 



Conjunction of Planets 631 

i.e. dark part depending on angular distance from 

sun. Thus the classifications are - 

Distance difference - (i) Madhya vrtta birhba- 
Average bimba size seen at average distance. 

(ii) Sphuta vrtta bimba - Current size of bimba 
depending on the sphuta distance of planet. 

Phase difference (iii) Madhya bhasvara bimba 
- Half lighted phase corresponding to about 90° 
angular separation from sun. 

(iv) Sphuta bhasvara bimba - True lighted 
portion according to angular separation. 

Actual bimba - (v) Drktulya bimba -' which is 
actually seen according to distance and phase 
effects. 

Verses 41-42 : Diameters of planets 

Diameters of star planets in yojana are 
Mangala (450), Budha (930), Guru (4750), Sukra 
(2600) Sani (3500). 

These divided by 2213 give the bimba in kala 
in sun orbit. 

Notes : (1) Yojana value in sun's orbit is 
converted to kala by dividing it with 2213 as 
explained in candragrahana (chapter 8) verse 25. 

le made by 1 yojana at that distance is 

: — radian = ='- — kala 

sun's mean distance mean distance 

as 1 radian = 3438 kala 




hi, 



I' A 

& ■ ■ ■ . 



3438 1 

kala = tttt kala exact. 






mi:i 



m v 



76,08,294 2213 

This exact value indicates, that distance of sun 
has been calculated on basis of this ratio, after the 
diameter of sun was assumed 72,000 yojahas 
according to Atharvaveda. 



632 Siddhanta Darpana 

All other text books have compared the 
diameters of planet in moon's orbit , but siddhanta 
darpana has compared them in sun's orbit. The 
linear diameter is based on accumption that the 
distances are inversely proportional to angular 
speed i.e. proportional to period of rotation. As 
the comparative distance of moon and sun on that 
basis was rejected due to correct looking value of 
Atharvaveda, reference to moon's orbit also was 
rejected. However, the distance of other planets 
and sun are considered proportional to their 
periods of rotation. This is justified because all 
planets move round the sun and moon around 
earth both according to siddhanta darpana and 
modern theory. 

Period of rotaiton T and distance D are not 
directly proportional, but according to Keplar's 
third law 

T 2 a D 3 

where D is distance (semi major axis) 

Thus T a D 3/2 instead of Ta D assumed here. 
Thus, actual relative distance of farther planets will 
be lower than calculated here. 

There is evidence in vedas that orbit was not 
meant the linear circle, but the surface of sphere 
on which this circle moves due to rotation of pata. 
The same concept is used in Jain texts also. If time 
period is considered proportioanl to volume, then 
this relation T 2 a D 3 holds as T a 4/3 II r 3 , D 

2 2 3 

4ot where r is radius of orbit. Then T and D 
both ax . Time volume relation is only a conjecture 



Conjunction of Planets 

(2) Comparison of values 

Mean angular diameters of planets (1) 



633 



Planet 


Aryabhata 


I Vatesvara 

• 


Tycho 


Siddhanta 


Modern 




and Lalla 




Brahe 


Darpana 


(mean) 


Mars 


1'15".6 


1"19".2 


1'40" 


8" 


14". 3 


Mercury 


2'6" 


2'12" 


2'0" 


25" 


9" 


Jupiter 


3'9" 


3' 18" 


. 2 # 45" 


25" 


41" 


Venus 


6'18" 


6'36" 


3'15" 


70" 


39" 


Saturn 


1'34" .5 


1'39" 


1'50" 


10" 


17" 


(2) 












Planet 


Old Surya 


Brahmagupta 


Surya 


Aryabhata 


Bhaskara II 




siddhanta 


and Sripati 


siddhanta & 
Bhattotpala 


n 




Mars 


4' 


4'46" 


2' 


4' 45" 


4'45" 


Mercury 


T 


6'14" 


3' 


6' 15" 


6'15" 


Jupiter 


8' 


7'22" 


3' 30" 


T 15" 


7'20" 


Venus 


9' 


9' 


4' 


9* 


9* 


Saturn 


5' 


5'24" 


2'30" 


5' 15" 


5' 20" 



Aryabhata and Vatesvara have reduced the 
values of surya siddhanta and made them more 
correct. They are generally more correct then values 
of Tycho Brahe, who had observed with telescope. 

Old surya siddhanta value is 2 to 2-1/2 times 
the values of modern surya siddhanta and have 
been approximately followed by others in table (2). 

Siddhanta Darpana has evidendy reduced the 
value of angular diameters in ratio of about 11, in 
which ratio the diameter and distance of sun have 
been increased. However, compared to surya 
siddhanta ratios, he has made increase in mercury 
and venus diameters and reduced the ratio of outer 
planets. For outer planets ratio of siddhanta 
darpana and modern values are Mars 1 : 1.8, Jupiter 
1/1.64 Saturn 1/1.7 

Ratio for inner planets is 

Merucry 2.8/1, venus 1.8/1 



634 



Siddhanta Darpana 



One reason may be that, the visibility of outer 
planets reduces due to large distance from sun, 
hence they appear smaller. 

Minute values of old S.S/Seconds value of 
siddhanta Darpana for outer planets is 

Mars 1:2, Jupiter 1:3 Saturn 1:2 

For inner planets 

Mercury 1:3.6, Venus 1:7.8 

It is quite probable that Candrasekhara has 
calculated the angular diameters of inner planets 
according to their average distance from sun which 
is much less. 

Comparison of linear diameters : 

Planet Siddhanta Darpana 

yojana Earth=l 

754.3 0.472 

601.6 0.376 

8324.5 5.203 

802.1 0.501 

14,776.4 9.235 

Surya siddhanta relative figure are almost 
correct for all planets except Jupiter and venus 
whose value is about half the true value. These 
errors might have come due to incorrect ratio of 
time period and distance. However, sun's diameter 
comes to be less than, jupiter and saturn also as 
it is taken only 6500 yojanas. 

Figure of siddhanta darpana are more correct 
with two errors - Jupiter and saturn values reduced 
by about l/4th of correct value, mars about half 
value. But mercury and venus vahes have been 
increased about 1.45 and 1.63 times the correct 
value. This appears to be due to error in estimating 
angular diameters of inner planets (more). 





yojana 


Earth * 1 


Mars 


450 


0.281 


Mercury 


930 


0.581 


Jupiter 


4750 


2.969 


Venus 


2600 


1.625 


Saturn 


3500 


2.188 



[odemdiam 


Distance 


Earth = 1 


Earth = 1 


0.536 


1.523688 


0.403 


0.387099 


10.925 


5.202803 


0.990 


0.723331 


9.01 


9.538843 



Conjunction of Planets 635 

Other correct feature is that all planets are 
assumed much smaller than sun (72,000 yojana 
diameter) due to more correct diameter of sun. 

Verses 43-44 : Madhya bimba 

The angular diameters of budha and sukra in 
sun orbit are their mean diameter. Angular 
diameters of other planets are obtained by multi- 
plying their angular bimba in sun orbit by sighra 
paridhi of the planet and dividing by 360°. The 
angular diameters in vikala are 

Mangala 8, budha 25, guru (25), Sukra 70 and 

sani 10 

Notes : Average distance of budha and sukra 
is same as average distance of sun from earth as 
these inner planets are in small orbit round sun. 
For outer planets 

Distance of sun _ Sighra paridhi of planet 
Distance of planet 360° 

as sun is the sighra kendra for outer planets. 
Hence the formula for angular diameters. 

Verses 45-50 : Bhasvara and sphuta bimba - 
Madhyama bimba (angular diameters) are kept at 
two places. At one place, it is multiplied by 
utkramajya in the process of fourth sighra phala 
and divided by two times trijya (6876) Result will 
be substracted from the madhyama bimba at other 
place. This will be bhasvara bimba of the planet. 
For budha and sukra, when their sighra is in 
6 rasis beginning from makara etc, then fourth 
phala kala and sighra koti kala are added; jya of 
the sum is added to trijya. Then 3 rasis are added 
to the sighra kendra of budha and sukra and its 



636 Siddhdnta Darpana 

koti rasi is substracted. Bhujajya of the remainder 
is multiplied by bimba diameter and divided by 
two times the trijya. Result will be bhasvara bimba 
of these two planets. If sighra phala kala and koti 
kala is more than 3 rasi together, then jya of the 
sum is added to trijya to find multipliers for budha 
and sukra. 

Thus we get madhya vrtta bimba and madhya 
bhasvara bimba. They are separately multiplied by 
trijya and divided by their sighra karna to get the 
sphuta bimba and sphuta bhasvara bimba. 

Notes : (1) Surya siddhanta has given the 
following formula; spasta bimba 

Madhyama bimba X 2 x T rijya 
Trijya + Fourth sighra karna 

This formula is correct if the madhyama bimba 
is calculated at distance of sun, but in surya 
siddhanta it is calculated at the distance of moon. 
However, this formula is correct for siddhanta 
daprana where madhya bimba has been calculated 
at sun's distance. This is the second formula given 
here and is based on the following ratio. 

Trijya + 4th s ighra karna 

— : Trijya 

= Madhya bimba : spasta bimba. 

More accurately this should be found from 
true distance of planet from earth i.e. 4th sighra 
karna instead of average of trijya and sighra karna. 
Thus siddhanta darpana gives correct formula for 
spasta bimba based on ratio — 

Sighra Karna : Trijya = Madhya bimba : spasta 
bimba 



Conjunction of Planets 637 

(2) Earlier correction : Bhasvara birhba is 
measure of relative visibility. If depends upon 
distance from sun due to which brightness 
decreases in ratio of square of distance (inverse 
square law). However, due to phase also the 
brightness increases and is more when angular 
distance between planet and sun is 90° to 270 °, 
we calculate utkramajya which deducted from trijya 
gives kotijya. Phase of a planet is equal to 
illuminated area divided by whole area of disc. 
The crecent GCHFG, bounded on one side by the 
semi ellipse GFH and on other side by semi circle 
GCH, is the illuminated part. GH is the line of 
cusps and CD the diameter perpendicular to it. 
Let CD = 2a. Then the phase 

Illuminated arc 

Area of disc 

G 



j-, ■: 





H ■ ■ E ^ To earth 

Figure Ba - Phase of the graha Figure 8b - Lighted part dueto sun 



£■ 



- n a (CM - FM) 



CF 



(1) 



n a 2 2a 

Now in figure 8b, hemisphere ACB is lighted 
sun and hemesphere CAD is seen from earth. 

£SME = d 

Hence CF = CM - FM 

= a - a cos AMC 



638 



Siddhanta Darpana 



= a (1 + cos EMS) = a (1+cos d) 

Hence phase = (1 + cos d)/2 (2) 

If we measure the difference between planet 
and sun, as d', then d' ■= 180* - d 

Hence phase = (1-cos d')/2 - - - (3) 
Thus in formula we find the utkrama jya 
R (1 cos d') and divide it by 2 R to get the phase 
according to eqn. (3). By substracting this portion 
from total bimba, we get the unlighted portion 
which is away from sun. 

For budha and sukra, the phase is calculated 
when they are on farther side of sun (sighra kendra 
270° to 90°) when they are more illuminated. We 
approximately find distance of mercury from 
superior conjunction (adding 3 rasi to 270 is ) 
or inferior conjunction. One gives illuminted figure 
but on farther side, the other gives dark portion 
but on nearer side. 

Verse 51 : When sighra kendra of budha and 
sukra is 6 rasi i.e. they are between earth and sun 
then they are like black holes compared to bright 
sun in its disc. 

Verses 52-55 : Now observed bimba of 
bhasvara is stated. Bhasvara bimba appear sthula 
(i e round without sharp cusps) like a candle flame 
at far distance (which appears a round point instead 
of elongated figure) 

When a bright object is very far, it appears 
215 times its real angular diameter. Bhasvara bimba 
kala is multiplied by 16 and square root of tne 
product is taken. That is observed value of seen 
bimba. 






m? 



Conjunction of Planets 639 

Notes : (1) Reasons of this arbitrary assump- 
tion are not known. However, from the discussions 
three variations in bimba emerge - 

Sphuta bimba is linear change in angular 
diameter which decreases with distance - like 
diameter of moon and sun. 

Bhasvara bimba is the lighted portion of disc 
due to its phases like moon. 

Observed bimba of a point like object is seen 
215 times bigger. But square root of bhasvara bimba 
is divided by 4 only for the diameter of observed 
bimba in kala. 

(2) Logic of this method is not under stood. 
A point like object will appear bigger due to 
diffraction or scattering of light. That increase in 
angular width will be fixed and not 215 times the 
radius. Its angular increase will be same for sun 
and moon also. Possibly Candrasekhar had seen 
some star planets with a telescope set at 215 times 
magnification as mentioned by Prof. J.C. Ray in 
his introduction. 

Verse 55 : Naksatras are self illuminated and 
their distance is fixed, as it is almost infinite 
compared to planetary distances. Still their seen 
angular diameter should be found out. 

Note : Though the stars are point like, two 
stars or star and a star planet are seen together, 
even when they are slightly separated. There are 
two reasons for that - 

Due to scattering of light in atmosphere, the 
point object appears to have, some width. 

Even when they are separated, their distance 
cannot be seen if it is less than limit of resolution 
Pf human eye. 



640 Siddhanta Darpana 

Verses 56-60 : Types of conjunction - 

Now types of conjunction (yuddha or 
samagama) are being stated. 

(1) When the observed birhba of two planets 
touch each other, that is called ullekha yuddha 
(touching conjunction). 

(2) When birhba of • a planet enters another 
planet, it is called vedha or bheda yuddha (piercing 
conjunction). 

(3) When north south difference of two 
planets in conjunction is less then sum of semi 
diamters, then it is arhsa vimarda yuddha (part 
eclipse conjunction). 

(4) When the mutual distance is more than 
sum of semi-diameters, then it is called apasavya 
(i.e. separated), 

Then the difference is upto 1° (60 kala) i.e. 
Distance between centres - sum of semi diameters 
< 60 Kala. 

(5) When the separation is more than 60 kala 
then it is called samagama. 

(6) When, in an apasavya (separation less than 
60 kala), one planet is bright and the other is dark 
(inferior planet between earth and sun), then it is 
called yuddha. 

When both are bright, it is called samagama 
When both are dark, it is called kuta yuddha. 

(7) When two planets are equal in longitude 
(i.e. in yuddha) and» northern planet has bigger 
diameter, then the southern planet is conquered. 

When both are equal, then north birhba is 
conquered, south is victor. 






Conjunction of Planets 641 

Sukra is victor, whether in north or south (as 
it has largest birhba among tara grahas and is 
brightest). 

Notes : These are only conventions for 
predicting ftirture events and described in Brhat 
sarhhita etc. Here samagama has been used twice. 
One is conjunction when rim distance is more than 
60 kala. Another is yuddha in which both planets 
are equally bright. However, conjunction of moon 
with a star has been called samagama generally. 

Verses 61-63 - South north distance 

To know the north south distance, two 
drkkarmas have already been described. As in 
eclipse, nata and larhbana corrections also are 
needed for the true north south distance. Earlier 
astronomers didn't observe or calculate less than 
1/2 degree or 30 kala, hence they ignored nata and 
lambana of tara graha which is much smaller. Still 
for academic interest it is being described to explain 
the mathematics. < 

Verses 64-67 : Nati of if ptaW£ 

Parama nati of sun is 22 vikala. Madhyama 
nati of budha and sukra als6 in same. Nati Kala 
of budha and sukra (22/60) is multiplied by trijya 
and divided by last slghra karna. Quotient is again 
multiplied by vitribha natamsa (drkksepa) and 
divided by trijya for spasta nati of budha and 
sukra. 

For other three planets (mangala, guru and 

sani, parama nati of ravi is multiplied by their 

| iighra paridhi and divided by 360. Quotient is 

niultiplied by trijya and divided by fourth slghra 



$42 Siddhdnta Darpana 

karna. Result is again multiplied by drkksepa and 
divided by trijya to get the spasta nati. 

As in solar eclipse, viksepa of the 5 planets 
is corrected with spasta nati to get the sphuta sara. 

Notes : (1) Average distance of budha and 
sukra is same as that of sun, hence their parama 
madhya nati will be same as that of sun. As the 
parallax reduces in proportion to distance similarly 
for outer planets - 

me an parallax of planet _ mean distance of sun 
mean parallax of sun " mean distance of planet 

sighra paridhi 
360° 
as sun is considered sighra kendra of outer 
planets. 

True parama nati mean distance 

(2} " = " 

■ . Nfean parama nati True distance 

Trijya 

Fourth sighra karna 

(3) Parama nati is for horizontal position for 
which drkksepa or jya of vertical distance (south) 
is maximum or equal to trijya (R). Since nati 
depends on jya of vertical distance towards south 

spasta nati 
parama nati 





(4) Correction of sara for nati has already been 
explained for solar eclipse-. They are added if in 
same direction and subtracted if in different 
direction. 



Conjunction of Planets 643 

Verses 68-71 : Laihbana correction 

At the time of conjunction, parama nati of 
the planet is multiplied by drggati and divided by 
trijya, and quotient is multiplied by jya of difference 
between planet with vitribha lagna and divided by 
trijya. Then we get sphuta laihbana (parallax in 
east west direction). 

When planet is east of vitribha lagna, sphuta 
laihbana is added to planet, otherwise substracted. 

After laihbana correction, some difference 
comes in the longitudes of the planets. Then again 
conjunction time is corrected when the longitudes 
are same. For this new conjunction time, again 
laihbana is calculated and, new conjunction time 
is found, when they will be equal in longitude. 
1 After repeated processes, we get the true conjunc- 
tion time. 

Notes : Parama nati of the planet is found as 
above section. Drgjya is the distance of planet from 
* vertical direction and nati will be proportional to 
it. Its value in ecliptic is proportionately known 
from distance of planet from vitribha. This has 
been explained in solar eclipse- 

Verses 72 : Conmjunction of graha and 
naksatra - 

| Since naksatras are very far from earth, their 

| speed and parallax both are zero. Hence, its 
| conjunction with a planet is calculated only from 
| ttie speed of graha. 

i 3feises 73-75 : Bheda yuddha 

' " Since lambana and nati are very difficult, this 

I correction is done only for rinding bheda yuddha, 



644 Siddhanta Darpana 

when bimba of one planet enters the bimba of 
another. For other conjunctions this is not neces- 
sary. 

Bheda of sun by budha or sukra should be 
calculated like other conjunctions. When they are 
moving in opposite direction (budha or sukra is 
vakri), then from sum of the gati and when both 
are margl, by difference of gati, we calculate the 
conjunction. According to the respective sizes of 
bimba, times of sparsa etc can be found. 

Sara of vakri budha or sukra is very little so 
vedha of sun is done by them. In this case time 
of sparsa etc is found from sum of speeds. 

Verses 76 : Moon and star planets - 

Moon is corrected for nati and lambana and 
its vedha by graha bimba is calculated like sun. 

Verses 77-90 - Samagama of moon and star 
planets- When a tara graha and candra have equal 
longitude (rasi, amsa and kala), then for finding 
their lambana, madhya gati of moon (790/35) is 
divided by 14. Quotient (56/28) is reduced by 
lambana of tara graha found from its parama nati. 
This will be maximum value of nati difference of 
moon and that planet. 

Parama nati difference is kept at two places. 
It is multiplied by 60 (to make it vikala) and divided 
by madhyama gati difference of moon and the 
planets. If the planet is vakri, then it is divided 
by sum of gati. This is time of parama lambana in 
ghati etc; It is multiplied by drggati of that time 
(vitribha sanku) and divided by trijya (3438). Result 
is made asu. It is assumed kala and its jya is called 
'para'. 









Conjunction of Planets 645 

Bhuja and koti jya of difference between moon 
and lagna is found. Difference of bhuja jya and 
para is squared and added to square of koti jya. 
Square root of the sum will be chaya karna. Kotijya 
is multiplied by para and divided by chaya karna. 
Result will be madhyama lambana. 

Madhyama lambana is multiplied by difference 
of madhyama gati and divided by difference of 
sphuta gati if the tara graha is margi. If tara graha 
is vakri, then madhya lambana is multiplied by 
sum of madhya gati and divided by sum of sphuta 
gati. Result is spasta lambana. 

This lambana is substracted from moon, if it 
is east (more) of vitribha lagna, otherwise added. 
Then the new time of conjunction is found when 
moon and graha have the same lipta. The lambana 
asu is multiplied by second vitribha sanku and 
divided by 1st vitribha sanku (before lambana 
correction). After correction of moon by this sphuta 
lambana asu, we find the sphuta madhya kala of 
conjunction. 

According to method of solar eclipse, 
drkksepa of vitribha lagna at mid conjunction time 
is found. Its 1/513 is added and divided by 61 to 
find nati of moon. 

By method of solar eclipce, from nata jya of 
vitribha lagna, sara and aksarhsa valana are found. 
When sara of moon and graha are in same 
direction, difference is taken, when they are in 
different direction they are added. This sara will 
be useful for diagram (parilekha) of samagama. 
When graha is south from moon, sara will be 
yamya, when it is north, sara will be saumya. 






646 Siddhanta Darpana 

For tara graha, moon is chadaka (eclipser) 
because it is closest to earth. Since moon has more 
speed, sparsa of its bimba by the planet will be 
in east and moksa will be in west. 

After doing ayana drkkarma of graha, graha 
and naksatra conjunction is calculated from nati 
corrected sara. 

Notes : The methods are exactly similar to 
methods of solar eclipse. Only difference is that 
the tara graha can be vakri also, when sum of gati 
is used instead of their difference. 

Verses 91-96 - Parilekha 

Like diagram of eclipse, we draw the 
manaikya vrtta (circle with radius as sum of semi 
diameters) inside khagola vrtta with radius 57 718' 
angula = 3438' radius. From same centre moon 
circle is drawn. For valana of khavrtta, sparsika 
valana in east and mauksika valana in west is given 
in their own directions. From valanagra, we draw 
a line to the centre of moon, called diksutra. 

From the points where diksutra cuts 
manaikya, we give sara at the time of sparsa and 
moksa in their direciton (north or south). The line 
from saragra points (end points of sara) to centre 
of moon, cuts the moon bimba on two points 
indicating entry and exit points of graha or 
naksatra. 

In conjunction of naksatra and moon, sanku 
of vitribha lagna is multiplied by 100 and divi<fcd 
by 231 to give jya of parama lambana or 'para'. 

Like moon and star/planet conjunction, vakri 
budha and sukra enter the sun disc from east side 



Conjunction of Planets 647 

and exit from west side. Since sun has no sara, 
the sara of only budha or sukra is the total sara 
and direction of this will be the direction of sara. 
pise of sun will be in centre of samasa vrtta (circle 
with radius as sum of semi diameters). 

Notes : The discription in parilekha, chapter 
10 is sufficient to understand this. 

Verses 97-106 : Observing shadow of planets 

From rays of star planets like mangala, we 
cannot see the shadow of a 12 angula sanku. 
Ilience, a mirror is kept on the shadow end point 
|ind sanku top is seen in mirror. Exactly at shadow 
pnd point, the planet and sanku end are seen in 
lone direction. 

On a plane level surface, we keep a vertical 
lanku of 5 hands hight. In it 12 divisions are 
marked, each being 1 angula. Sanku will be strong 
md straight and its surface will be cylinderical. 

As explained in Triprasnadhikara, from the 
iiata kala of the planet at desired time, we find 
the shadow length of 12 angula sanku. With that 
semi diameter a circle is drawn with sanku centre 
m the centre. Direction points are marked (earlier 
m day time) and from the centre, lines are drawn 
m east west and north south direction. 

Then the kranti jya of graha at the desired 
time is multiplied by chaya karna and divided by 
lambajya. Quotient will be karna vrttagra in angula. 
It will be substracted from palabha for north kranti 
"Si the planet and added for south kranti to get 
ija of shadow in angula (its distance in north 
south direction from sanku). On north south 

through centre, we mark a point at distance 





648 



Siddhdnta Darpana 



of chayagra bhuja in the opposite direction of 
inclination of the planet. From shadow length 
(chaya) square, we substract the square of chaya 
bhuja and take the square root. Result is dharatala 
sanku which is called koti also. When planet is in 
west half of sky, koti is given east from the end 
point of chaya bhuja. At the point of shadow circle 
where it cuts, shadow end will lie. At this point 
a tube will be kept in direction of the sanku top 
and we see from below. Or a mirror is kept and 
its reflection is seen. 



N 



W 



jr N 1 


^S p 




f N 2 


* 1 


I 





E W 




Figure 9 (a) Figure 9 (b) 

Notes - In figure (a) ENWS is circle with 
radius of shadow length. Current direction of 
shadow is OP. OP is length of shaodw, ONi is 
chaya bhuja, NiP is its koti. Hence OP 2 = ONi 2 + 
NiP 2 . ON2' is palabha, i.e. shaodw at the time of 
equinox midday. The difference with bhuja is karna 
vrttagra, N1N2' = ONi - ON 2 

N1N2 - 7 Sin d' where 6' = spasta kranti, 

cos <p r . . / 

<p = aksamsa. This is explained in 
Triprasnadhikara 



Conjunction of Planets 649 

In figure (b) OC is sanku of 12 angula length. 
OP is chaya and PC is chaya karna. Thus PC is 
in direction of planet at G. If we keep a tube in 
PC direction, planet can be seen from P end. By 
a long dark tube we can see a planet in day time 
also as scattered day light is absorbed by inner 
surface of tube and only light of planet is seen 
which is not obstructed. Alternatively, by keeping 
a horizontal mirror at P we can see planet by 
keeping eye in direction of PK, Here PK makes 
same angle with vertical PC as LCPC = d 1 in 
opposite direction. 

Verses 107-108 : Seeing the yuti 

At the time of yuti (conjunction of planets) 
we keep two sanku at the distance of sara difference 
and from the same point P we can see both planets 
through tube or a mirror. Result of different types 
of yuti are given in books of samhita (like Br hat 
samhita of Varahamihira). 

Verse 109 : Increased size of vrtta bimba - 

Here, the bimba of planets decribed or birhbas 
of stars to be told later, are very bright, hence 
they are seen 16 times more lighted than moon. 
At the time of sunrise and sunset, their discs are 
as bright as moon, hence their bimba value has 
been stated as 4 times = vT6 larger. Thus the real 
t angular diameter is 1/4 of the seen diameter. 

Notes : (1) This explains the logic of formula 
! for observed bimba in verse 54. But it is not correct. 

(2) Due to diffraction of light, two points at 
angle less than 6 radians cannot be seen separately 
where 



650 Siddhdnta Darpana 

Sin e ' ~5~ 

where D is diameter of aperture through 
which planet is seen (it may be aperture of pupil 
of eye or lens of a telescope). A is wave length of 
light (4000 to 8000 angstrom = 10* cm units). This 
is Raleigh criterion. Thus for visible light, when 
pupil is 1.5 mm diameter in day time, we cannot 
see two points which are separated by les than 
about 1' kala. In night time when pupil is bigger 
it will be about 20" vikala. Thus the angular 
diameters of outer planets are smaller than the 
limit of resolution of eye and even when they are 
separated, they appear together. This explains as 
to why separation upto 1 kala is called samagama 
and only for larger separation, they are really seen 
separate. 

Thus at the time of conjunction, the effective 
diameters of planets are seen digger. 

' (3) Other reasons of fluctuation are scattering 
of light, and fluctuations in atmosphere, which are 
almost same for both the nearby stars or planets. 
The stars are so distant, that their angular diameter 
is zero even after seeing through largest telescopes. 
Their diameter of conjunction time is seen much 
more than 215 times due to diffraction. 

Verse 110 : Solar eclipse due to sukra 

To find eclipse of sun due to venus, their 
bimba and size of other tara graha is stated. In 
kali year 4975 (1874 AD) there was a solar eclipse 
due to sukra in vrscika rasi (i.e. in Nov.- Dec. 
month). Then sukra bimba was seen as 1/32 of 
solar bimba which is equal to 650 yojana. Thus it 



r 



Conjunction of Planets 651 

|5 well proved that bimba of sukra and planets is 
much smaller then sun. 

Verses 111-112 : Prayer and conclusion 

May Lord Jagannatha remove our ignorance, 
who defeats beauty of blue clouds by his blue light 
aiid lives on sea coast. 

Thus ends the eleventh chapter describing 
conjunction of planets in Siddhanta Darpana 
written for tallying calculation and observation and 
education of students by Sri Candrasekhara, born 
|ii famous royal family of Orissa. 



&■■■ 




Chapter - 12 

CONJUNCTION WITH STARS 

Verse 1 - Scope - To know the conjunction 
of planets with naksatras, the longitude and 
latitude of identifying star in each naksatra starting 
with asvinl, shape of naksatras and number of stars 
in it and bimba of yogatara (identifying star) is 
stated first. 

Verses 2-11 : Longitudes and latitudes of 
identifying stars (yogtara) 



S.No. 


Name 


Beginning 


Name of 


Longitude 


Latitude Position 


of 


of 


point 


yogatara 


of 


of 


of 


Naksatra 


naksatra 


longitude 




yogatara 


yogatara 


yoga tar 
a 


1. 


Asvinl 


, 0' 


/?Arietis 


10*07' 


+8*29' 


10*07' 


2. 


Bharani 


13 "20* 


Arietis 


24*21 


+1027 


11*01 


3. 


Krttika 

• 


26'4(T 


rfTauri 


3608 


+403 


928 


4. 


Rohini 

• 


40*0' 


«Tauri 


4556 


-528 


556 


* 5. 


Mrgasira 


53*2(y 


A Ononis 


4951 


-1323 


631 


6. 


Ardra 


66*40* 


erOrionis 


6454 


-1602 


-146 


7. 


Punarvasu 


80 '0' 


/JGeminorum 


8922 


+641 


922 


8. 


Pusya 


93*20' 


<5Cancri 


10452 


+005 


1132 


9. 


Aslesa 


106 # 40' 


crCancri 


10947 


-505 


307 


10. 


Magna 


120*00' 


aLeonis 


12558 


+028 


558 


11. 


Purva 
Phalguni 


133*20' 


£Leonis 


13727 


+1420 


407 


12. 


Uttara 
Phalguni 


146*40' 


/JLeonis 


14746 


+1216 


106 


13. 


Hasta 


160*0' 


<JCorvi 


16936 


-1212 


936 


14. 


Citra 


173*20' 


aVirginis 


17959 


-203 


639 


15. 


Svati 


186*4(r 


aBootis 


18023 


+3046 


-617 


16. 


Visakha 


200*0' 


aLibra 


20114 


+020 


113 


17. 


Anuradha 


213*20' 


<$Scorpii 


21843 


-159 


523 


18. 


Jyestha 


226*4C 


aScorpii 


22554 


-434 


-046 


19. 


Mula 


240*0' 


AScorpii 


24044 


-1347 


044 


20. 


Purva 
Asadha 


253*20' 


dSagittarii 


25043 


-628 


-237 



Conjunction with Stars 



653 



2i. Uttara 

Asadha 

22. Sravana 

23. Dhanistha 

24. Satabhisaj 

25. Purva 
bhadrapada 

26. Uttara 
bhadrapada 

27. Revafi 



266* 4C oSagittarii 25832 -327 



•0' 

• in' 



280*0 
293*20 
306 '4C 

320*0' 

333*20' 
346*40' 



aAquilae 

^Delphini 

AAquarii 

aPegasi 

yPegasi 

£Piscium 



27755 
29229 
31743 
32938 

34518 

35601 



+2918 

+3155 

-023 

+1924 

+1236 

-013 



-808 

-205 

-051 

1103 

938 

1158 

921 



These are the modern positions and names 
of identifying stars. Nirayana longitude of Citra 
(a-Virginis) was fixed as 180° at 285 AD to fix the 
nirayana position accurately in zero ayanamsa year. 
Now it has become 179° 59' due to negative proper 
motion of citra. 

Verse 12-24 : Verses 12-14 give the number ' 
of stars in each naksatra. Verses 15-18 give the 
shape of each naksatra. 

Verses 19-22 give the direction of yogatara 
within the naksatra (this can be known from their 
latitude and position in naksatra also given in 
previous table). Verses 23-24 give the diameter of 
yoga tar a in vikala. Actually the diameters are 
almost zero even by talescope viewing, they are 
measures of visual magnitudes of brightness. The 
yogatara positions of 28 naksatras including Abhijit 
according to siddhanta darpana in previous verses 
and the other details are given in chart form. 



SI. Naksatra 
No. 

1 Asvini 

2 Bharani 

3 Krttika 



Owner 
(yajurveda) 

Asvina 

Yama 
Agni 



Yogatara Latitude Bimba Shape No. 
Longitude Vikala of 

Stars 



9*45' 

2100 
3515 



+10*30' 

+1100 
+415 



2 
3 



Horse 3 
mouth 

Triangle 3 

Flame 6 



654 



Siddhdnta Darpana 



4 


Rohini 

• 


Prajapati 


4630 


-537 


7 


Cart 
(Sakata) 


5 


5 


Mrgasira 


Soma 


6015 


-1330 


2 


Cafs 

paw 

or 

head 

of dear 


3 


6 


Ardra 


Rudra 


6500 


-1540 


7 


Coral 

or 
water 
drop 


1 


7 


Punarvasu 


Aditi 


9015 


+630 


8 


Bow 


5 


8 


Pusya 


Brhaspati 


10400 


+115 


2 


Arrow 


3 


9 


Aslesa 

■ 


Sarpa 


10800 


-1200 


4 


Dog 

tail 


5 


10 


Magha 


Pitr 

* 


12600 


+022 


6 


Plough 


5 


11 


Purva 
Phalguni 


Bhaga 


14300 


+1200 


12 


weight 

on 

two 

ends 

of 

beam 


2 


12 


Uttara 
Phalguni 


Aryama 


15300 


+1300 


13 


-do- 

• 


2 


13 


Hasta 


Savitr 


16500 


-1100 


4 


Hand 


5 


14 


Citra 


Tvasta 

■ ■ 


17900 


-210 


7 


Pearl 


1 


15 


Svati 


Viyu 


19300 


+3300 


13 


Coral 
orjewel 


1 


16 


VisSkha 


Indragni 


20700 


-200 


2 


Shed 
or tent 


5 


17 


Anuradhd 


Mitra 


21830 


-200 


4 


Snake 
hood 


7 


18 


JyesthS 


Indra 


22530 


-415 


7 


Teeth 
of Boer 


3 


19 


Mula 


Nirrti 

* 


24040 


-1330 


5 


Cronch 

or 

lion's 

tail 


9 


20 


Purva 


Apah 


25000 


-630 


4 


tusk 


4 


* 


Asadha 

• * 














21 


Uttara 
Asadha 


Vi§vcdavah 


25630 


-340 


4 


Chute 
(Supa) 


4 




Abhijit 


Brahma 


25630 


+6200 


14 


Triangle 

or fire 

bail 


3 



Conjunction with Stars 



655 



22 


Sravana 


Visnu 


27300 


+3000 


7 


Arrow 

or 
short 
men 


3 


23 


Dhanistha 


Vasava 


28530 


+3600 


3 


Long 
drum 


5 


24 


Satabhisaj 


Varuna 


31745 


-020 


3 


Canopy 


100 


25 


Purva 
bhadrapada 


Aja-Ekapada 


32200 


+3200 


4 


Cot or 

weights 

from 

beam 


2 


26 

27. 


uttara 
bhadrapada 

Revati 


Ahirbudhnya 

Pusa 

* 


33800 
0*00' 


+2800 
+500 


4 
3 


do 
drum 

rtr fi«Vi 


2 
32 



Notes : (1) Yoga tara in north position of 
naksatra - (1) Asvini 5. Mrgasira 11. PQrvaphalgunl, 
16. Visakha 20 Purvasadha 21. Uttarasadha, 25 
Purva Bhadrapada, 26 - Uttara bhadrapada 

Yogatara in centre - 19. Jyestha, 22 Sravana, 
17 Anuradha, 3. Krttika, 8 Pusya 

Yogatara in isana (north east) - 7 Punarvasu, 
13- Hasta, 19-Mula. 

Yogatara in west - 23. Dhanistha, - Abhijit 

Yogatara in east - 4. Rohini, 9. Aslesa 

Yogatara in south - 10. Magna (very bright), 
27. Revati 12. Uttara phalguni 

Yogatara in agni kona (south east) - 24. 
Satabhisaj. Single stars are in 6. Ardra, 14. Citra 
and 15. Svati, hence there is no difference between 
the naksatra and yogatara. 

(2) Shape . of naksatras have been decribed 
dMferently by different authorities. Actually, it is 
only imagination and convention. 

(3) Longitudes and latitudes also differ slightly 
according to different authorities. 



656 Siddhanta Darpana 

(4) It may be seen that many yagataraa do 
not come within extent of their naksatra. Hence 
three naksatras are divided into purva and uttara 
part. In unequal division of naksatras, most of the 
naksatras have yogatara in their extent. 

(5) It has already been stated that diffraction 
and partly scattering of light in atmlsphere spreads 
the point like stars. Bright star has bigger spread 
as, greater spread of diffraction ring remains 
visible. 

Verses 25-40 - Other stars - 

Now many other stars are described. 

(1) Lubdhaka (Sirius) - It is brightest star south 
of punarvasu with birhba of 20 vikala, dhruva 77° 
and dhruva prota kranti 40 ° . Surya siddhanta name 
of this star is lubdhaka. Bhaskara II has given its 
longitude (polar) as 86°. It is 8.6 light years away 
and brightest star. 

(2) Mrgavyadha - There is another small star 
south of punarvasu. Surya siddhanta and Lalla 
have called this same as lubdhaka, but it is different 
star. Its dhruva is 56°, south sara 32° and birhba 
is 10 vikala It may be identified with Orion, which 
is also called hunter is greek stories borrowed from 
Egypt. 

Its south latitude is same as south latitude of 
Magadaskara (now Malagasi) an island in south 
east direction of Africa - hence this island was 
called Mrga or Harina dvipa 

(3) flvala - This is a group of three stars 
between mrgavyadha and ardra. Its middle star is 



Conjunction with Stars 657 

yogatara, whose dhruva is 61° and south sara 
23°30'. 

(4) Hutabhuk - According to siirya siddhanta, 
its dhruva is 52° and north sara is 8\ 

(5) Brahmahrdaya - According to siirya 
siddhanta, its dhruva is 52° and north sara 30°. 

(6) Prajapati - It is 5° east brahmahrdaya 
whose dhruva is 57° and north sara is 38*. (Surya 
,siddhanta) 

Modern observations have indicated the fol- 
lowing positions {by author). 

(4) Hutabhuk - Dhruva 58° 15', sara 5° 15' north 
birhba 6" vikala 

(5) Brahmahrdaya - Dhruva 56°, north sara 
23°, birhba 16" 

(1) Lubdhaka is now called prajapati. 

(7) Apamvatsa - This is 5* north from citra. 

(8) Apa - This is 6* north of Apamvatsa. It 
is also called apyavasu. 

Dhruva of both (7) and (8) above are equal 
to citra. North sara of (7) is 2 - 50* and (8) is 8°5(r. 

(9) Agastya - Its dhruva is 95" and south sara 
is 75°. Its dhruva becomes sphuta after doing 
ayanamsa correction. Its bimba is 18* vikala. 

(10) Yama - Its dhruva is 22% sara is 66* 
south and birhba is 8". 

Surya siddhanta has stated dhruva of agastya 
as 90°. This was the value at the time of writing 
that book when 121 years were remaining in satya 
yuga. In Kali era 4251, Bhaskara II has stated its 



658 Siddhdnta Darpana 

dhruva to be 87°. He has stated dhruva of 
punarvasu as 93° and Agastya 6° less i.e. 87°. At 
the time of siddhanta darpana, it is 17°30' west 
from punarvasu i.e. 90 "IS 7 - 17° 30' = 72° 45'. From 
agastya dhruva 95% on substracting ayanamsa 22 *, 
we get the same value 73° approximately. The 
change of agastya dhruva from 87° at the time of 
siddhanta siromani when ayanamsa was 11° 30' to 
mithun 13° (73 °) is the change in 719 years (1869 
AD). 

♦Notes : Ayanamsa correction is not needed 
when the distances have always been measured 
with respect to fixed stars. There may be some 
error in identification of stars. Otherwise relative 
motion of stars is very little and negligible 
compared to ayana movement. Opinions differ 
regarding correct identifications of these stars with 
current greek names used. Modern names of 
yogatara have already been given. Agastya is 
canopus, apamvatsa is 0-virginis and Apa - <5 vir- 
ginis, Agni or hutabhuk in p tauri. Prajapati is 
ft aurigae, Brahma is a aurigae. 

Verses 41-56 - Saptarsi mandala 

Since saptarsi mandala (great bear) is moving, 
its dhruva has not been stated by earlier 
astronomers. Still, I state their position, based on 
my experience. 

In north direction saptarsi mandala spread in 
east west direction like a bullock cart is very 
prominent in the sky. It has been most reverred 
in sarhhita and purana. 



Conjunction with Stars 659 

Within this group, there is an upward raised 
line towards east. Marici is in its front. Behind it 
Vasistha is with Arundhati. Still west from Vasistha 
is Angira. 

After that, is a quadrilateral. In its isana kona 
(north east), lies Atri. South from it is Pulastya 
and west from Pulastya in Pulaha. North of Pulaha 
is Kratu. The great circle joining Pulaha and Kratu, 
cuts ecliptic in some point, the naksatra or rasi of 
that point is considered the rasi of saptarsi. 

At present Pulaha and Kratu are in 21° of 
sirhha i.e. 3rd quarter or purva-phalguni. 13 
kalamsa east from them is Pulastya. 

Atri is 5 kalarhsa east from Pulaha, 9 kalamsa 
east from Atri is Angira, 8 kalamsa east from Angira 
lies Vasistha and 8 kalamsa east from Vasistha is 
Marici. 

Arundhati is a very small star, east from 
Vasistha which is barely visible and can be seen 
with telescope. This is not giver of good or bad 
omen, like the seven main stars. Its birhba is 1 
vikala. Birhba of Atri is 3 vikala, and all others are 
8 vikala. Mutual distance between these stars is 
same and equal to 10 pala kalam'sa. 

This 10 pala is multiplied by 1800 and divided 
by rising time of that rasi at equator. The quotient 
is added to the dhruva of Pulaha or Kratu (sirhha 
21° = 141°). We get the dhruva in rasi etc for 
other stars. East west angular distance (along 
ecliptic) of saptarsi is 43°, but due to its position 
in sayana kanya and tula, it appears 46° (in rising 
time at equator). 



660 Siddhania Darpana 

Distance from ecliptic along dhruva prota vrtta 
(great circle through dhruva, not kadamba ,- pole 
of ecliptic) in north direction are - 

Kratu 56°, Pulaha 51', Pulastya 53", Atri 59' 
Angira 60*, Va&stha 62° and Maria 60°. 

If sara of Vasisttha from kranti is fixed, then 
in the end of even quadrant, it wil be 4* from 
dhruva (North sara 62° + kranti at end of even 
quadrant 24° = 86' i.e. 4° from dhruva at 90°). 
Even if sphuta kranti of saptarsi remains same, 
their sara changes with change in rasi. 






P 



a 



u " x ^ - - 

8 



a 



m * s 



Polasis s v 
P' s 



Horizon 



N 



a, ;' 



S. 



*> North Pole 

a Aksamsa 
North 



Figure 1 Position of saptarsi and pole star 

, Notes (1) Due to earth's rotation, saptarsi 
makes a revolution around north pole in direction 
of line pa of its western stars. Three positions at 
3 hour intervals are shown from east to west. 
Polaris P' is very close to north pole (58' Kala 
distance) and is called pole star. P is <p angle above 
north horizon, where is local north aksamsa. The 
stars are indicated by greek letters starting from 



Stars 


Greek nam 


1 Marici 


rj Alkaid 

r 


2 Vasistha 


£ Mizar 


3 Angira 


e Alioth 


4 Atri 


6 Megrez 


5 Pulastya 


y Phad 


6 Fulaha 


p Merak 


7 Kratu 


a Dubhe 



Conjunction with Stars 661 

western lower star. Siddhanta counts them from 
eastern end. Modern names, distances and visual 
magnitudes are given below - 

Visual magnitude Distance in 

light years 
1.87 210 

2.06 88 

1.79 68 

3.3 

2.44 90 

2.37 78 

1.81 107 

More visual magnitude indicates lesser bright- 
ness, thus Atri is least bright and farthest/ hence 
its bimba vikala has been indicated small. 
Arundhati is a small star, below Vasistha called 
Alcor (magnitude 5). Mizar (Vasistha) itself is a 
double star when seen from telescope. It appears 
that Atri has faded now but earlier, it was equally 
bright. 

(2) Mythology : Callisto was attendent of 
goddess Juno but was more beautiful. To protect 
her from jealosy, Callisto was turned into bear by 
god Jupiter. When her son Areas, thought her a 
bear and wanted to kill her, he was also turned 
into bear (ursa minor) 

According to Puranas, Saptarsis are mental 
sons of Brahma. There is a separate set of sap tarsi 
for each of 14 manu periods of which 7 are yet to 
come. Rsi and Rksa have been used for star, sage 
or bear also. Hence ursa in Persian means saint, 
in Greek it means bear. Like this bear around north 



662 Siddhanta Darpana 

pole, Russian bear exists. Russian was Rsika and 
it is land of bear. Proverbially Russia is called 
Russian bear. Rsi denoted sage and bear as both 
had long hairs. Hence the name great bear came. 

(3) Motion of Saptarsis : Only Vatesvara 
siddhanta chapter 1 verse 15 has given the number 
of revolutions of saptarsi which is 1692 in a yuga. 

On that basis, Karana sara of Vatesvara has 
given a method to calculate movement of saptarsis, 
as quoted by Albirun! (India I, page 392) - 

Multiply the basis (i.e. years elapsed since 
beginning of saka 821) by 47 and add 68000 to the 
product. Divide the sum by 10,000. Quotient is 
position of saptarasi in rasis etc. 

According to this formula, saptarsi has a 
motion of 47 signs per 10,000 years which is 
equivalent to 1692 revolution in 43,20,000 years, as 
stated above. 

The position in saka year 821 (Kali year 4000 
was) 

1692 x 12 x 4000 . A . . , 68,000 

signs = 1 revolution + innnrt signs 

43,20,000 * B 10,000 

This accounts for the addition of 68000 in 
formula. 

(4) The stars of the constallation of the saptarsi 
do not have a motion relative to naksatras. So the 
statement of revolution is not correct. This appears 
to be the reason why many standard astromers 
like Aryabhata, Brahmagupta, Sripati, Bhaskaras I 
and II, Surya siddhanta etc do not deal with the 
subject at all, as being outside the pale of 
astronomy. Therefore, Kamalakara was constrained 



I 






Conjunction with Stars 

to say in his Siddhanta Tattva Viveka, Bhagraha 
yutyadhikara, verses 25-36 - 

''Sage Sakalya has given the motion of the 
sages with their positions in his time. Surya and 
others who explain the nature of the celestial 
sphere in their works do not give it, and therefore, 
the theory cannot be sustained astronomically. 
Even today, this motion mentioned in the samhitas 
is not observed by astronomers. Therefore, the 
seven real sages who are the presiding deities (of 
these stars) are only to be supposed to be moving 
unobserved by men, for the prediction of the fruits, 

thereof." 

But the motion has been accepted as a fact 
by certain common people and authors of the 
Puranas, and an era called Laukika era by the 
peopie of Kashmir region and saptarsi era by the 
puranas have been founded on this theory. 

Mahabharata mentions, that when Yudhisthira 
ascended the throne, Saptarsis were in magha 
naksatra. 

Vayu purana chapter 99, tells that saptarsi's 
remain for hundred years in one naksatra. Hence 
they complete the round of 27 naksatras in 2700 
divya years. However, same purana chapter 57, 
t" tells that saptarsi naksatra is of 3030 human years. 
Hence human year appears to be taken as 12 sideral 
revolution of moon. Divya year here means 1 solar 
year. 

2700 solar years = 2700 X 365.256263 days for 
sidereal years 



664 Siddhanta Darpana 

2700 x 365.256263 , 
= 12 x 29.321661 lunar ^ sidereal 
= 3007.968 years 

Varaha Mihira has written in Brhatsamhita 
13/3, that according to Vrddha Garga, Saptarsis 
were in Magna in the rule of Yudhisthira. Raja 
Tarangini of Kalahana has followed this era only 
in writing ancient history. 

(5) Explanation : Kamalaakara has explained 
that it has no relation with astronomy and it is 
only for astrological predictions. Siddhanta Darpana 
has tried to justify the movement of saptarsis on 
basis of their measurement of position on dhruva 
prota vrtta on ecliptic. Normally kadambaprota 
vrtta is used for ecliptic and dhruva prota for 
equator. This causes difference and has explained 
the difference in terms of ayanamsa. His calculation 
of difference from Bhaskaracarya time is based on 
ayana - movement. However, this will have a cycle 
of 26000 years and not of 2700 years of saptarsi 
era. Hence Candrasekhara has not mentioned the 
saptarsi era but has vaguely tried to justify its 
movement. 

My explaination is based on basis of vedanga 
jyotista which was current in Mahabharata period, 
Rk jyotisa has a cycle of 19 years in which 5 years 
are of samvatsara type (starting between Magha 
sukla 1 to Magha sukla 6.) Yajus jyotis starts with 
5 year cycle of 366 days each, but this also becomes 
equivalent to 19 year cycle with 6 ksaya samvatsara 
in 5 cycles of 5 years each. P.V. Holey has assumed 
a bigger yuga of 19 X 8 + 8 = 160 years because 



(Conjunction with Stars 665 

it gives very little error, but has not explained the 
mechanism of arranging the last eight years. This 
is not corroborated any internal evidence in the 
text. 

However, saptarsi era was very much in use 
and was accepted in the calendar system. This 
appears to be based on system of naming a century 
(100 solar years) on a naksatra in same way as we 
name every guru varsa or other solar years on 
basis of 1st week day of the civil year. Thus, we 
can name the century on basis of first naksatra of 
moon (or may be sun) in a century, if the vedic 
yuga system is followed. Then bigger yuga should 
be 19 X 5 + 5 years = 100 years instead of 160 = 
19X8+8 years. Thus in a century, we can take last 
5 years as the first 5 years of the 19 years Rk cycle 
Or 5 years first cycle of yajusa jyotisa. In taking 
yajusa cycle, the 19 year cycle doesn't break and 
in sixth cycle first 5 years are subcycle which make 
a century. It can also be seen that all cycles of 19 
year start with Sravistha but after five years of 
yajus cycle, sixth year starts with satabhisaj which 
is the next star. Thus on completion of 100 years, 
|ln this calender, moon gains one naksatra in this 
jpfcalertder. Thus each sucessive century will start 
pWfch one naksatra later, which will be saptarsi 
jlaksatra or naksatra of the century. 

Sri Holey has opined that Rk jyotisa was 

itely written before 2884 B.C. According to 

traditions vedic texts or puranas were written 

300 years of Mahabharata war. Thus it 

y indicates the calender system fixed in 

dhisthira time who had really started an era. 



!?,•• 




666 Siddhanta Darpana 

That time year started with Magha sukla paksa, 
hence saptarsis were assumed to be in magha to 
start with. 

(6) Mutual distance between stars of saptarsi 
mandala is not equal as stated here. 

Verses 57-59 : Kranti of circumpolar stars 

Sphuta kranti of yama and Agastya is always 
same. Kranti of naksatras starting with Asvini keeps 
changing. Kranti of saptarsi mandala is fixed 
according to some, but changes according to others. 

Earlier astronomers have assumed motion of 
saptarsi mandala as 8 kala per year from east to 
west. But I (author) have not seen such gati. Hence 
I do not agree to it. 

Ayana sanskara of yama, Agastya and saptarsi 
mandala has been instructed to be done in opposite 
direction. This is valid for author's time only (1869 
AD). 

Notes (1) Ayana samskara is needed because, 
here dhruvamsa have been given for yogataras and 
other stars. As stated earlier, Candrasekhara has 
assumed oscillatory motion of ayana, and according 
to this, the present backward movement will 
change after 2200 AD. 

(2) 8 Kala movement of saptarsi is 800 Kala 
or 1 naksatra in a century whcih has been stated 
by Vatesvara. This has not been accepted by author 
correctly. 

(3) Circumpolar stars are near dhruva (pole 
star) and appear to move round it. This is true 
for south polar region also. This depends on local 
latitude of the place. Day length of a planet or 



Conjunction with Stars 667 

t r increases by carajya which is increase in half 
day length for north kranti. It is decrease for 
southern kranti. 

Increase in half day = Carajya 

= R tan d tan <p (<3 =spasta kranti, <p= latitude) 

If this is equal or greater than R, then the 
increase is equal to half day of equator itself and 
there will be no night i.e. the star will never set. 

For this tan <f> tan <5 > I 

Thus for any star with north kranti, 

it is circum polar, if 

tan 6 > cot <p 

Similarly for south kranti, also, if 

Tan d > cot <p 

then the star will never rise. 

Verses 60-62 : North pole star (dhruva tara) 
has birhba of 4 vikala. This is not the real position 
of dhruva i.e. pole of equator and dhruva prota 
is not drawn through it. The seen dhruva tara is 
1°24' away from surface centre of equator. Hence, 
when revati naksatra comes on meridian i.e. 
dhruvatara at beginning of mesa, appears 84 Kala 
above the pole. 

When sravana, punarvasu naksatra are on 
yamyottara, dhruva rises above horizon equal to 
local aksamsa. Again, when citra naksatra comes 
on yamyottara, dhruva in 84 kala nata from its 
kendra. 

Notes : Polaris ( Ursae minoris) is a star of 
second magnitude and is 58' Kala away from 
celestial pole in west direction in 1950 AD. Celestial 



668 



Siddhdnta Darpana 



pole is moving towards polaris upto 2105 AD when 
it will be only 30' away, then will begin to recede 
from it. 

It may be seen that Draco or dragon group 
is pole of ecliptic i.e. pole of solar system. Thus 



Polasis 1900 AD 
a 




8400 AD 



/ 2700 BC 



4600 BC 



Hercults 



a'Ly 



14800 AD 



Figure 2 - centre of this circle is ecliptic pole 

proverbially sun as visnu is under draco or 
sesanaga with 3-1/2 turns. Base of human body 
cakras is also called serpent of 3-1/2 turns 
(Kundalini). Thus it was called draco is Chaldia 
and dragon of 3-1/2 turns in China also. 

Since pole star is 84 kala from pole to mesa 
Q , or revati naksatra, it appears above north pole 
when revati is on meridian and below 84 kala when 
naksatra 180° opposite citra is on meridian. When 
naksatras 90° from these are on meridian, (Sravana 
or punarvasu), altitude of north star will be same 
as pole (though east or west by 84 kala). 

Verse 63 - Similarly south pole star also 
appears to move around south pole like a bullock 
rotating the oilseed crusher in a circle. 



Conjunction with Stars 



669 



Note - There is no conspicous star near south 
pole. Octans group contains south pole, %ut its 
brightest star v is of 3.7 magnitude and official 
pole star a (sigma octanis) has 5.5 magnitude. It 
is in line with bigger arm of south crpSs group. 

VersesN>4-66 : Three measurements - Three 
angular measurements (west to east) are used - 
ManSrhsa, Kalamsa and Ksetramsa. Rising times of 
rasls being different, Ksetramsa and Kalamsa are 
different. After one revolution, both complete 360 \ 
Manamsa and Kalamsa are same on equator, but 
difference between them increases as we go further 
from equator, in north or south direction. '.' 

Notes : The different measures depend on 
different system of coordinates shown in figure 3. 




P = 

QE 
K = 



Figure 3 - System of Coordinates 
Celestial Pole (Dhruva) 

= Celestial equator* * 

Pole of equator (Kadamba) 



g 7 Siddhanta Darpana 

YL' = Plane of the ecliptic 

Y= First point of sayana mesa (vernal equinox) 

L,L' = First points of sayana makara and 
karka/winter and summer solstice). 

S = a heavenly body 

PS = A great circle through P,S, cutting 
equator at Q and ecliptic at B. 

Y Q = Right ascension = a = Kalamsa (time 
is measured along equator rotation) 1 Kala at 
equator = 1 asu, R.A. of 1 hour = 15° at equator. 

QS = Declination = Kranti 6 

KS = Great circle through K, S, cutting ecliptic 

at C. 

Y C = Celestial longitude = A = Ksetramsa or 

bhogamsa 

CS - Celestial latitude = p = Sara or viksepa 

Y B = Polar longitude or dhruvaka = 1 

BS = Polar latitude = viksepa (dhruva) = d 

Polar longitude (dhruvarhsa) and latitude 
(viksepa) have been used only in this chapter to 
indicate position of stars as we observe them with 
reference to fix position of pole. 

WE' is a circle parallel to equator in north at 
ange d, latitude of the circle Y. M = <f> where M 
is point on P corresponding to mesa 0°. Absolute 
length of arc between M and corresponding 
position Q' is almost same as great circle ^between 
them. The great circle between M and Q' lS 
manarhsa 

Arc MD = Arc Y Q. cos 6 



Conjunction with Stars 671 

Though the angular difference between MQ' 
and YQ is same, manamsa is less. It becomes 
lesser, if d increases i.e. we go farther from equator. 

Kalamsa is the distance along equator, hence 
it is equal to rising time of rasis (at equator). 

Verses 66-68 : Saptarsi measures 

Stars in saptarsi are taken from east to west 
along declining longitude (desantara), not in north 
south direction (aksamsa) (Second half of verse 66) 

Here difference between dhruva viksepa of 
pulaha and kratu is only 5°. Similarly, east west 
difference manamsa between kratu and marici is 
25° and difference in viksepa is only 0°30\ Sphuta 
kranti can be calculated. 

Verses 69-70 : Conversion of three measures 

Sphuta kranti and dyujya are calculated. Sum 
of two dyujya for kranti vrtta and visuva vrtta is 
made half - it is called hara. 

Manamsa multiplied by trijya and divided by 
hara gives ksetramsa (degree on ecliptic) 

Ksetramsa multiplied by rising time of its rasi 
and divided by 180° gives kalamsa. This way 
kalamsa of marici and kratu can be found. By 
reverse process, manamsa can be found from this 

^sa. 

Notes : (1) In notes of previous section, vide 
e (3) manamsa is measured along WE' parallel 

*p equator which is diurnal circle of a star of this 

on. It is easier to convert it to kalamsa as 

ined in the note, or in spastadhikara for 

Sacculation of day length. 

Manamsa M = H cos <5 






672 Siddhdnta Darpana 

(where H is kalamsa, d is kranti) 

H. R cos d _ H Dyujya 

R Trijya 

However, d here is measured along dhruva 
prota SQ instead of SC line. Thus length along 
ecliptic is reduced due to lesser rising time of B 
compared to C, and increases due to oblique length 
of B. Thus instead of dyujya we take average 
dyujya and trijya. 

M Trijya 



Kalamsa H = 



-(dyujya + trijya) 



(2) Ksetramsa is converted to kalamsa as per 
the following approximate ratio used for calculation 
of lagna - 

Rising time of rasi Rising time for ksetramsa 
Rasi (1800 kala) " Kalamsa in kala 

Rising time of ksetramsa is measured along 
equator, hence its asu is equal to kala of kalamsa. 

Verses 71-75 : Variation of kalamsa and 
manamsa- Dhruva star moves in a circle of 360°, 
hence its manamsa 

360 X 84 otAOtt 

= — = 8'48" 

3438 

i.e. 1° of this circle is equal to length of 8'48" 
on equator. 

Due to change in kranti, if shape of sapktarsis 
remains fixed, then with change in kranti, their 
rising time will also change with change in dyujya. 

Since kalantara of saptarsis is fixed with 
change in kranti then, east west distance will 



Conjunction with Stars 673 

change with change in aksamsa. If kranti is fixed, 
then kalamsa will be constant. 

If ksetramsa is constant, then kalamsa and 
manamsa wil vary. Like kalamsa and bhagamsa, 
relation between kalamsa and manamsa also can 
be found. 

Verses 76-79 : Sara of naksatras 

The dhruvamsa of naksatras given here are 
already with ay ana drkkarma. Their sara also is 
sphuta i.e. in dhruva prota vrtta. 

But sara of graha is asphuta, i.e. in kadamba 
prota vrtta. After drkkarma, it will become sara in 
direction of dhruva prota vrtta of stars. 

Even when the. kadmba prota sara of naksatras 
is same, their kranti in dhruva prota circle will be 
different due to east west difference. Hence the 
length of their day and night will be different (as 
the semi diameter of diurnal circle . - dyujya, 
depends on distance from equator in dhruva prota 
direction). 

If the sphuta kranti of a naksatra is more than 
the co-latitude of a place, the naksatra will be 
always rising at that place (it will be always seen 
there above horizon). If the south sphuta kranti is 
more than the co4atitude of the place, it. will never 
rise above horizon, i.e. always set. 

Notes : Dhruvamsa has already been ex- 
plained. This has been used for indicating position 
of stars because it is easier to observe them with 
reference to north pole. 

Circumpolar stars have already been ex- 
plained. For them, carajya = R tan d tan <l> is 



674 Siddhanta Darpana 

bigger than R, radius of equator, hence day length 
will be more than day night value. 

Thus tan d tan <p > 1 
or tan 6 > cot <p = tan (90° - <p) 
Here, d is kranti, O is aksamsa, hence 90 ° 
-O is lambarhsa. 

Thus <5> 90° - O 

Then the star will be always rising if kranti 
is bigger than lambarhsa. 

Similarly for south kranti, if carajya is bigger 
than R, day length wil be less than 0, i.e. the star 
will not rise. This means the same condition. 

Physically, we can understand it, because 
north pole is above horizon at angle equal to local 
aksamsa. Distance from north pole to the star is 
90° - d which should be always less thanO if the 
star is to remain above horizon. Thus <p> 90* - 
d or 6 > 90° - O 

i.e. Kranti > lambamsa 

Similarly, south pole is 0° below south 
horizon, A star with south kranti will be 90° - d 
away from south pole. If this distance 90° - d is 
less than O , then the star will never rise. 

* 
> 

Verses 80-84 - Conjunction of graha and 
naksatra 

Ayana drkkarma is done for the involved 
graha and difference of dhruvarhsa of graha and 
naksatra is found. The difference in kala is divided 
by sphuta gati of the graha in kala to get result 
in day, ghati etc. If dhruva of graha is less than 
naksatra, the conjunction will occur after that 
interval, if it is more, then the yoga has already 



Conjunction with Stars $75 

occurred, that period before. When graha is vakri 
(retrograde) then opposite order will happen (i.e. 
if graha dhruva is less, conjunction has happened 
earlier). For this conjunction time, we again find 
difference in sphuta dhruva of graha and naksatra 
and get the more accurate value of conjunction 
time. After successive approximations, we get the 
correct conjunction time. 

After that, sphuta kranti and cara of graha 

and naksatra are found and cara is calculated. That 

; will give periods of their day and night. From that 

| we get the values of udaya and asta lagna of graha 

I and naksatra for their rising and setting times. As 

\ explained before, the rising and setting times will 

j be when sphuta sun reaches those positions (of 

I udaya and asta lagna). By finding difference of 

arhsa at rising setting times, we get the proportional 

\ difference between graha and naksatra according 

!' to the natakala (aksa drkkarma explained earlier) 

and again we revise the conjunction time, when 

longitude of graha and naksatra are same after 

aksa drkkarma. 

* 

As explained in conjunction of planets, we 

I find the north south difference between graha and 

naksatra from difference of their dhruva prota sara. 

I Distance between discs is obtained by substracting 

I the semi diameter of bimbas from this distance. 

I Notes : The methods of ayana and aksa 

fdrkkarma have already been explained in conjunc- 
tion of planets. Here, the problem is simpler, 
because the position of naksatra is alrady stated 
corrected by ayana drkkarma. Further, we need 
*u>t calculate motion of naksatra, because they are 
gfixed. Here also conjunctions will be different 
Recording to distance between discs. 



$76 Siddhanta Darpana 

Verses 85-87 : Bheda of naksatras 

Planets can enter the following 13 naksatras 
(or do 'bheda' in their extent) - 

Rohinl, pusya, krttika, citra, magna, punar- 
vasu, anuradha, jyestha, visakha, revatl, satabhisaj, 
purvasadha, and uttarasadha. 

Other fifteen naksatras are never crossed by 
planets (no bheda) - 

Asvinl, bharani, mrgasira, ardra, aslesa, purva 
phalguni, uttaraphalgunl, hasta, svati, mula, ab- 
hijit, sravana, dhanistha, purva bhadra pada and 
uttara bhadrapada. 

Among crossed (bhedya) naksatras, punarvasu 
is crossed by every planet. Purvasadha, revatT, and 
krttika are sometimes crossed. Others are less 
frequently crossed according to kranti of the graha. 

The planet whose south kranti in 14th degree 
of vrsa (44°) is more than 2° 20', can cross the 
rohini (in shape of sakata - cart). 

When other naksatras are pierced or entered 
by graha, it is confirmed by seeing with instrument. 

Shapes of naksatras and planets moving north 
or south (beyond naksatra) can be seen in 
Brhatsamhita by Varahamihira. 

Thus positions of many stars have been told 
which are famous since ancient times. There are 
many other stars in unlimited number. Nothing 
has been told here about naksatras expect asvini 
etc. 

Notes (1) The graha move in ecliptic with little 
deviation according to their small sara. Many 
naksatras have large deviations, where the graha 



Conjunction with Stars 677 

will never reach. Punarvasu is lying on ecliptic, 
hence it is crossed by all planets. This was the 
naksatra which determined start of solar year and 
malamasa in lunar year in vedic era. Hence its 
name is punarvasu, i.e. resettle or restart of year. 
13 naksatras with less deviations can be crossed 
by planets. 

Sakata bheda - Rohini is in shape of cart i.e. 
sakata, hence its bheda is called sakata bheda. Its 
yogatara has 5° 32' south sara, but northern most 
star has 2° 35' south sara. According to siddhanta 
darpana it is 2° 20'. Moon has sara up to 5° hence 
it can easily cross rohini, but except budha and 
sukra, no other graha has parama sara of this 
value. Parama sara of sani is 2°29'39" hence sanis 
sakata bheda also appears impossible. But 
Varahamihira and Grahalaghava author have stated 
that sakata bheda by sani or mangala is very 
inauspicious. 

For siddhanta darpana value of sanis sara, its 
sakata bheda is just possible (at 2° 20' south sara). 

Mangala parama sara is only 1°51' according 
to siddhanta darpana and modern value but still 
less according to earlier texts. Vedha of rohini is 
possible only when south sara is assumed less, 
which is not given in the texts. 

According to star catelogues 3000 to 6000 stars 
only can be seen with naked eye. There are about 
10 11 stars in our galaxy (of average size of sun) 
and there are about 10 9 ) galaxies in universe. 

Verses 88-92 : Milky way - 

A fine circular way of dense fine stars in seen 
in the sky. This is called chaya patha, vaisvanara 



678 Siddhdnta Darpana 

patha or abhijit marga (akasa ganga also). It is 
proposed to describe it. 

This chaya patha is circular. It crosses ecliptic 
in sayana karka and sayana makara beginning. 
Again it extends 60° north from sayana mesa to 
63° south from sayana tula. This crosses south part 
of punarvasu and goes southwards. Then it crosses 
mula and sravana naksatras and goes upto centre 
of abhijit and sravana. From there, it goes 
northwards. From beginning point of karka, it goes 
north in two branches. This can be easily shown 
by diagram on a sphere. But in sky, it is seen half 
only at a time, hence it is impossible to show it. 

We can easily see the stars (separately) of 
milky way with telescope. We can also see pusya 
naksatra, black spots on sun, water, mountains 
and trees on moon. Telescope can show phases of 
budha and sukra also like moon. Ring around sani 
and many new planets and satelites can be seen 
with it. 

Notes : (1) The galaxy is called akasa ganga, 
chayapatha, visnupada etc. However vaisvanara 
patha is name of ecliptic according to many. The 
akasa ganga, is the disc portion of galaxy which 
is dense area with more number of stars, hence it 
looks like a way. The main portion of the galaxy 
is a disc of about 30 kilo per sec width. It has two 
spiral arms and sun is located in inner arm as 
shown in figure 4(a) and 4(b). Sun is 10 Kpc away 
from centre i.e. about 2/3rd of the radius. 1 persec 
= 3.26 light years approximately, kilo = 1000. The 
galaxy rotates along the central plane of disc, which 
is almost parallel to orbits of solar system, central 



Conjunction with Stars 



679 



portion rotates with uniform velocity almost like a 
rigid body. Stars in vicinity of sun in the disc are 
rotating with speed of about 220-250 kms/sec 
around galactic centre. Mass of galaxy inside sun's 
orbit is 1.4 X 10 11 sun masses. Total energy of 
galaxy also is about 0.8 X 10 11 of sun. Mass of 
stars is 2X10 44 gram. 




\« 



30 Kilopersec 




Figure 4a - Structure of galaxy Figure 4b - Spiral arms 

The points in figure 4a represent some of the 
globular clustures. The position of sun is marked 
with the sign. Regions are marked 1 to 5 - 1. The 
spherical subsystem, 2 - the disk, 3 -' the nucleus, 
4 - the layer of gas dust clouds, 5 - the corona. 
Radius of corona is at least a dozen time the radius 
of galaxy. 

Figure 4(b) is disc of the galaxy. The nucleus 
is at centre C. Two spiral arms are spreading from 
it. Sun is in one of the arms. ' 

Spread of galaxy can be 
seen from figure 4c. C is centre 
and E is edge of disc. Sun is 
S. So SC = 10, CE = 15 kpc. 
^.ESC=0 is spread of disc. 
Tan0 = 3/4 hence = 60° ap- 
proximately. Figure 4C - spread of galaxy disc 




680 Siddhdnta Darpana 

Central portion and the disc are dense and 
obscured by gases. It can be observed only by 
radio telescopes. It is believed that nucleus of 
galaxy contains huge black hole. Spherical subsys- 
tem contains old stars and globular clustures. They 
rotate with about l/5th velocity of disc stars. Mass 
of corona is many times the mass of galaxy, but 
its density is much less and it does not emit any 
light. It is felt only by its gravitation. 

Centre of galaxy is in mula naksatra. its old 
name was mula barhani - i.e. the root from which 
cosmic egg has spread. Probably its position as 
galectic centre was known. Linga purana also states 
that brahma travelled for 30,000 years in cosmic 
siva linga - this is the distance in light years from 
centre to sun. ' . ■ 

(2) A note about magnitude of stars - The 
visual magnitude of the stars has been made in a 
logarithmic scale. Star of 1st magnitude is 100 times 
bright than 6th magnitude i.e. increase of 5 
magnitude reduces the brightness by 1/100. Change 
in magnitude of 1 reduces the brightness by (100)' 1/5 
= 1/2.5 approx. Brightness in this scale is 

Sun - 26.5 i.e. 6,31,000 times moon 

Moon - 12.0 

Venus - 3.0 

Sirius -1.4 

Rohini + 1.0 Brahmahrdaya 0.1 

Absolute magnitude is measured by emitting 
power compared to sun in similar logarithmic scale. 



Conjunction with Stars 681 

Verses 93-94 : Prayer and conclusion 

May supreme lord Jagannatha destroy our 
forest of mishaps, who puts the golden ornaments 
to shame with his yellow dress, who is closely 
watching the creation and events in the cosmic 
egg, who is expert in dancing on hoods of Kaliya 
naga and who is radiant near tree of desires. 

Thus ends the twelfth chapter describing 
conjunction of graha and naksatra in siddhanta 
darpana written as text book and correction of 
calculation by Sri Candrasekhara born in famous 
royal family of Orissa. 




Chapter - 13 

RISING SETTING OF PLANETS, STARS 

Graharksodayasta samaya varnana 

Verse 1 - Scope - Now I describe the rising 
and setting of planets and stars. In sphutadhikara, 
udaya and asta have been roughly described on 
the basis of difference of their kendramsa. 

Verses 2-3 : Types of rising and setting - 

Udaya and asta are of two types - Nitya (daily) 
and naimittika (occasional or seen). 

In first nitya type, due to rotation of pravaha. 
(daily rotation of earth), planets and stars rise daily 
in the east and set in the west. Hence, it is called 
nitya (regular) or pratyaha (daily). 

The planets rise when they are far from sun 
and are visible. They set when they become 
invisible due to closeness of sun. This is called 
naimittika (i.e. casual) udayasta. 

Verses 4-6 : Rising setting of planets (Surya 
siddhanta) - Mangala, guru and sani, set in west 
when their longitude (rasi etc.) is more than sun, 
when it is less then sun, they rise in the east. 
Vakri budha and sukra also set in west and rise 
in east, when their longitude is more than sun or 
less than it. 

Here more and less do not mean numerically 
bigger rasi. If the planet is ahead of sun in nearer 
portion of arc, then it is more in rasi and if behind, 



Rising Setting of Planets, Stars 683 

it is west. For example, sun in mesa will be 
considered more than a planet in mina, because 
mesa is unmedately after mina. From mesa to mina 
direction, mina is greater in numbers but it comes 
at the end of circle. 

(Surya siddhanta) - When budha and sukra, 
moon are less than surya, they set in east. When 
they are more than surya, they rise in west. This 
is because budha and sukra are faster than sun. 

Notes : In general the rising and setting of 
planets etc is due to daily rotation of earth, due 
to which each star rises in east and sets in west. 
This is called daily rising and setting. Siddhanta 
darpana assumes that earth is fixed and the sky 
is rotated east to west by a wind pravaha, which 
is equivalent to daily rotation of earth. 

This chapters deals with the other type of 
rising and setting caused by brightness of sun. In 
western astronomy, it is called heliacal rising and 
setting (heliacal = caused by sun, Helios = sun in 
Greek). In this rising, when the planets are very 
close to sun and they rise around sunrise in east 
and set with sun, they cannot be seen due to 
closeness of sun. They are said asta (naimitika) or 
heliacally set. When they are slightly away from 
sun and are seen slightly before sun rise (in east 
or west) or after sunset, they are considered 
heliacally risen or naimittika udaya. 

First part of the discussion is about mangala, 
guru and sani which are slower than sun. When 
sun is behind them sun appears to be moving 
towards them. When they become very close, trlese 
planets become invisible. Before that closeness, they 



684 Siddhanta Darpana 

are seen after sunset in west. After some days, 
they become invisible due to closeness of sun, 
hence they are said to set (heliacally) in west. After 
the time of closeness, sun goes ahead, then the 
planets are seen in east before sun rise. Hence the 
three planets are said to rise in east (heliacally). 

When vakri budha and sukra are ahead of 
sun, then they are seen in west after sunset and 
set there itself. After some days, they go to the 
other side of sun (less longitude, or west) and they 
are seen in east before sunrise. Hence vakri budha 
and sukra set in west and rise in east. 

When margi budha, sukra (and candra) are 
behind sun, they become nearer due to more speed 
and become invisible due to closeness. Then they 
are behind sun and are invisible in east before sun 
rise. Hence they heliacally set in east when they 
go ahead of sun, they are visible in west after sun 
set and are said to rise in west. 

Verse 7-11 : Drkkarma for rising and setting - 

For knowing the rising or setting time of a 
graha in west, on the approximate day of rising 
or setting, spasta surya and graha are found at 
sunset time. If the rising or setting is to be 
calculated in the east, then on apporoximate day 
they are calculated at sunrise time. (The ap- 
proximate time of rising or setting is known from 
rough kendrarhsa as explained in spasta-dhikara). 
After that drkkrama of both the planets is done. 
(Surya siddhanta quotation). Ayana drkkarma is 
done first, then aksa drkkarma is done. 

Method for aksa drkkrama - Sphuta sara of 
graha in kala is multiplied by palabha and divided 



Rising Setting of Planets, Stars 685 

by 12. Result is multiplied by 1800 and divided by 
lagna asu of that time. Result will be in kala etc. 
This will added for south sara of graha and 
substracted for north sara when sun is in east 
horizon. When sun is setting in west, reverse order 
will be followed. 

(Surya siddhanta) Difference of aksa karma 
corrected graha and sun in asu is divided by 60 
to find kalamsa. For rising and setting in west, we 
find the difference between (6 rasis added to graha) 
and the sun. By correcting graha with that, we 
again find kalamsa difference. 

Nati correction in sphuta sara of moon is done 
by the method explained in surya grahana (chapter 
9). To see the setting of moon in east, it is added 
to accurate moon and substracted from it to see 

the rise in west. 

p 




N 



A 
Figure 1 - Aksakarma at Ksitija 

Notes - (1) Aksa drkkrama at Ksitija - In Fig.l 

NALS - Eastern horizon 
N = North point 
S = position of rising graha 
L = Udaya lagna 



656 Siddhdnta Darpana 

K = Kadamba, P = pole 

B = position of rising planet S on kranti vrtta 

C = Planet S on kranti vrtta on dhruva prota 
circle. 

ABD = Diurnal circle of B 

CL = Aksa drkkarma of C. 

Here the planet with south sara, rises after 
its ecliptic position B has risen, or dhruvaprota 
position of C has gone further above. 

Diurnal circle of B cuts, dhruva prota of S on 
D. 

ADS is a spherical right angled triangle, 
because AD is parallel to equator, hence perpen- 
dicular to dhruvaprota line. Hence L DAS = 
lambamsa = 90°^>, where is aksamsa, /.DSA=0 

Small triangle DSA can be considered a plane 
figure 

* Hence 

DA = Sin Z. DSA _ Sin <i> _ Pabbha 

DS ~ Sin L DAS " Sin (90° - <J>) 12 

Now, approximately SD = SB and DA = CL 

CL _ DA palabha 
BS ~ DS = 12 

^ Sara x palabha 
or CL = ^ , as BS = Sara 

Since in south sara the planet is above horizon 
at sun rise time, its aksa correction is added to 
the planet. 

(2) Kalamsa is the time before sunrise when 
a planet rises. It is equal to 1 asu for 1 kala 
difference on equator. The difference between sun 



Rising Setting of Planets, Stars 687 

and planet corrected for aksa karma wil be rising 
time difference in asu = Kalantara in kala 

, , . . . , Kalantara in kala 
Hence, kalantara in amsa = — 

Rising time diff in asu 

= '~ 60 

(3) Nati samskara is needed only for moon as 
it is very little for other planets. 

Verse 12 - When the larhbana corrected moon 
is at 11° kalamsa from sun, then it is seen on 
horizon. Wh?h its distance is less than 11 kalamsa 
it cannot be seen. 

Verses 13-16 - Rising of stars 

Sara of naksatra are 'bigger. Hence sphuta 
kranti is found from their sara. For this sphuta 
kranti, carajya and day length in asu is found. 
That will give daily rising or setting time and lagna. 
At the time of rising (or setting), we get the 
difference of lagna and sun. The rising time for 
that difference in asu divided by 60 will give 
kalamsa. 

The kalamsa at the time of rise in east or 
setting in west is only dependent on sun motion, 
because stars don't move. Hence, they rise or set 
at distance of kalamsa from sun in east or west 
like farther planets mangala etc. 

Before setting in west the stars rise in east, 
due to daily motion. It is not connected to distance 
from sun. 

Notes : The method explained earlier for 
grahas was approximate for small sara. But 
naksatras have bigger sara and accurate method as 



688 Siddhanta Darpana 

explained in chapter 11 - for conjunction of planets 
is to be used. The rising time difference is found 
by sphuta kranti of star and sun i.e. it will be 
difference in their carajya only. Since it is at times 
of sun set or sun rise, proportional difference for 
natamsa of sun need not be made. 

_. , . Carjya difference in asu 
Thus, kalamsa = — 

Verses 17-24 - Kalamsa of stars 

Kalamsa of naksatras in degrees depends on 
their birhba diameters in vikala. The observed 
values of kalamsa for successive rise in birhba vikala 
is given below - 



Birhba Vikala 




Kalamsa 


1 




24 


1/15 




23 


1/30 




22 


1/45 




21 


2 




20 


2/30 




19 


3 




18 


4 




17 


5 




17 


6 




16 


7 




16 


8 




15 


9 




15 


,10 




15 


11, 12, 


13 


14 


15, 16 




13 


17, 18, 


19 


12 


20, 21, 


22 


11 



Rising Setting of Planets, Stars 689 

23 to 28 10 

29 to 40 9 

Notes : Bimba value is not the diameter of 
stars because it is so small, it cannot be measured 
even with telescope. It is only a measure of 
brightness estimated empirically. Actually the 
visibility distance (kalamsa) from sun is one of the 
measures of brightness - expressed as bimba 
diameter. 

Verses 25 - Kalamsa of tara grahas 

Kalamsa of taragrahas are 

Sukra 9, vrhaspati - 11, budha 13, sani 15, 
mangala 17. 

For sukra and budha, the above values are 
average. Their kalamsa at cakra or cakrardha (0° 
from sun - farther side is cakra, 180° from sun i.e. 
near side is cakrardha) are 

Cakra Cakrardha 

Budha 14 12 

Sukra 10 8 

Note : At the end of cakra, on farther side 
of sun, the planets are farther hence light intensity 
is smaller. Hence, they become invisible at greater 
distance. Then they rise in east and set in west. 
At cakrardha, budha and sukra are between earth 
and sun and vakri, then they set in east and rise 
in west. 

Verse 27-29 : Rules for heliacal rising 

When difference between sun and the planet 
or star is more than the kalamsa, it will not be 
visible (due to light of sun). 



590 Siddhanta Darpana 

Difference of graha or naksatra with sun being 
less than its kalamsa in west, means it has already 
set. If difference in more than kalamsa, then it will 
set in near future. 

If in east direction, rising will be in reverse 
order. If difference (kalantara) is more than 
kalamsa, then planet has already risen, if less than 
kalamsa, it is yet to rise. 

Notes - (1) Condition of rising are 

(1) Planet should be above horizon. 

(2) It should be night time for its being visible. 
Even in night, slightly before sunrise or after 
sunset, it becomes invisible due to twilight. In the 
limiting case of rising in east, its difference from 
sun should be more than kalamsa. Bigger or 
brighter planet will be visible at lesser distance 
from sun. 

(2) Setting in west - slow planets mangala, 
guru or sani or stars are overtaken by faster sun. 
In west, they rise after sunset at the minimum 
distance of kalamsa, when it has become sufficiently 
dark. It the limiting case, they are east from sun 
at kalamsa distance, when distance is more, it will 
be reduced in future, when the planets or star will 
set. Same happens with vakrl budha or sukra. 

For rising at sunrise time, they should rise 
before sunrise, i.e. west from sun at kalamsa. 
Distance of sun increases due to its faster speed 
in east direction. Hence if it is more than kalamsa, 
it was equal to kalamsa earlier, when planet or 
star has risen. Vakrl budha and sukra also are 
separated further as they are in west and moving 
further west. 



Rising Setting of Planets, Stars 691 

Verses 30-33 - Day of rising or setting of 
planets - Graha and sun are added with ayanamsa. 
For setting time, six rasi is added to both. For 
graha and sayana sun at rising time (6 rasi added 
to each for setting time) rising time of their rasis 
are multiplied by daily speed and divided by (1800). 
Result will be kala gati at the time of rising or 
setting. 

We calculate the difference between rising 
times of the planet and sun at sunset or sun rise 
times before aksakarma. This is ista kalamsa from 

* * 

which kalamsa of rising is substracted. This 
kalantara is divided by difference of kalagati of sun 
and forward moving graha. If graha is retrograde 
kalantara is divided by sum of kalagatis. Result 
will be past days or coming days of rising or 
setting, as per rules explained earlier. 

Notes : (1) Since inclination of planetary orbits 
with ecliptic is very small, it can be assumed to 
move on ecliptic. 

1800 kala on ecliptic = rising time of that rasi 
on equator is asu 

~_, ^ . ^ , gati kala x rising time 

or Kala gati in 1 day = - - - (1) 

6 J 1800 v } 

(Kala gati in kala.) 

(2) Kalamsa antara or kalantara 

= Kalamsa of graha on ista day - parama 
jfcalamsa of graha 

i Past or remaining days 

%■ Kalantara 

= — (2) 

v Sun kala gati ± graha kalagati v ' 

,, Here + sign is for vakri graha and - ve sign 
*s for forward graha. (3) 



692 Siddhanta Darpana 

Verses 34-37 - Aksa drkkarma for stars. 

Difference of ista kalamsa between naksatra 
and sphuta ravi and the parama kalamsa is divided 
by kala gati of sun at the place of naksatra dhruva. 
This will give past time or remaining time for udaya 
kala (if sphuta sun at rising time is taken) or asta 
kala. 

Half day of naksatra is calculated for sphuta 
kranti and asphuta kranti. Their difference in asu 
is multiplied by 1800 and divided by rising time 
of the rasi for rising in east (or rising time of 6 
rasi + naksatra for setting in west).' 

Result will be aksa drkkrama in kala. When 
naksatra has north sara, aksa kala will be added 
to naksatra in west and substracted from naksatra 
in east. For south sara of naksatra, aksa kala is 
added to naksatra in east and substracted for 
naksatra in west. Result is drkkarma corrected 
dhruva and ksetrarhsa is found from that. 

Notes : (1) Naksatras have no proper motion, 
hence their rising time is calculated only from sun's 
motions. Here, in place of sun gati ± naksatra gati 
= sun gati - = sun gati only. 

Similarly, kranti of naksatra is fixed, hence 
ayana drkkarma is not done, only aksa karma is 
done. 

(2) Udaya lagna or udaya vilagna of a star is 
that point of the ecliptic which rises in the eastern 
horizon simultaneously with the star and the asta 
vilgana or asta lagna of a star is the point of ecliptic 
which rises on the east horizon when the star sets 
on western horizon. 



Rising Setting of Planets, Stars 693 

z 




:.jc 



Figure 2 - Aksa drkkarma of a naksatra 

Figure 2 is celestial sphere for a place of 
latitude O., SEN is horizon, S, E, N being south, 
east and north cardinal points. Z is zenith. X is 
position of star when it rises on the horizon 
(eastern). TEB is the equator and P its north pole. 
TLA is the ecliptic and L is the point of the .ecliptic 
which rises with star X i.e. star's udaya lagna. The 
point T where the ecliptic intersects the equator is 
the first point of Aries (sayana mesa). PXAB is the 
hour circle (dhruvaprota circle) of star X and A 
the point where it intersects the ecliptic. U is the 
point where diurnal circle of A intersects the 
horizon. 

Now arc EB is the ascensional difference 
(carajya) due to true declination (arc XB) i.e. spasta 
kranti of star. Arc EM is carjya due to madhya 
Ikranti (of the star's position on ecliptic) i.e. arc AB 
of the star. The difference of these carajya is arc 
MB in asu. In asu of arc MB, portion CA of ecliptic 
nses. CA has been approximately considered equal 
to LA, aksa drkkrama of star. 



694 Siddhanta Darpana 

Thus aksa drkkrama = Carajya for spasta 
kranti - carajya for madhya kranti. 

Since dinardha = 15 ghatl + carajya 

difference of dinardha = diff. of carajya - (1) 

= aksa drkkarma 

Longitude of the star's udaya lagna L i.e. arc 
TL 

= arc TA - arc LA = arc TA - arc CA approx 

= Polar longitude - aksa drkkarma. (2) 

This explains when the star is north of the 
ecliptic, then aksa drkkarma is substracted from 
star to find udaya lagna in east. 

For asta lagna, it will be added to polar 
longitude of star (dhumvamsa) added to six rasis. 

(3) Rules for rising and setting can be stated 
as star rises heliacally when 

sun's longitude = udaya lagna of star + 
kalamsa. It sets heliacally when 

sun's longitude = astalagna - kalamsa - 6 signs. 

Star is invisible if, 

Sun's longitude - udaya lagna < kalamsa 

or, asta lagna - sun's longitude < kalamsa + 
6 rasis 

This can be stated in terms of udayarka and 
astarka. Udayarka (or udaya surya) is position of 
sun when a star rises heliacally. 

Astarka is position of sun when a star sets 
heliacally. 

Calculation of udayarka — Star's udaya lagna 
is taken as sun's longitude and it is assumed that 



Rising Setting of Planets, Stars 695 

time elapsed since sunrise is equal to star's kalamsa 
ghatis. Lagna at that time is itself udayarka. 

Calculation of astarka - Star's asta lagna is 
taken as sun's longitude and kalamsa ghatis of 
time before sunrise, we calculate the lagna. By 
adding 6 rasis to this lagna we get astarka. 

Theorem (1) - If astarka > udayarka, star will 
never set. 

When sun = udayarka, the star rises heliacally. 
Thereafter, as the sun moves, distance of sun from 
udayalagna increases and star remains visible. Since 
astarka > udayarka, the same happens when sun 
= astarka. 

The star therefore, does not set when sun = 
astarka. Thus setting is impossible in this case. 

This happens, when star has sufficiently big 
north latitude (for places of north latitude), such 
that star's aksa drkkarma > star's kalamsa (on 
ecliptic). For, in the case. 

Udayarka = star's polar longitude - aksa 
drkkarma + kalamsa 

< Star's polar longitude 

and, astarka = star's polar longitude + aksa 
drkkarma - kalamsa 

> star's polar longitude 

So that, Astarka > star's polar longitude 
> udayarka 

Theorem (2) - If, for a star, astarka 

< udayarka, then the star will rise and also set. 
The star will remain set when, astarka < sun 

< udayarka and will remain visible when sun 
< astarka but > udayarka. 



696 Siddhdnta Darpana 

Proof - When sun = astarka, the star sets 
heliacally. As the sun's longitude increases, the 
distance between asta lagna and sun diminishes 
and star remains heliacally set. This happens until 
sun = udayarka, when the star rises helically. Thus 
sun remains set until, astarka < sun < udayarka. 

When sun goes beyond this limit, it is helically 
visible. 

This case happens when the star's latitude is 
north and its aksa drkkarma < kalamsa of star. 

For, udayarka = polar longitude of star - aksa 
drkkarma + kalamsa on ecliptic 

> star's polar longitude 

and, Astarka = Star's polar longitude + aksa 
drkkarma - kalamsa 

< star's polar longitude 

so that, astarka < star's polar longitude 

< udayarka 

This also happens, when star's latitude is 
south. 

For, udayarka = star's polar longitude + aksa 
drkkarma + kalamsa 

* 

> star's polar longitude 

Astarka = star's polar longitude - aksa 
drkkarma - kalamsa 

< star's polar longitude 

so that, Astarka < star's polar longitude 

< udayarka. 

Rule for set period : A star remains heliacally 
set until astarka < sun < udayarka 



Rising Setting of Planets, Stars 697 

Between this period we take sun's speed as 
the speed at position of star's polar longitude which 
is in between these two values, hence can be taken 
as average speed. Hence this period for setting 
Udayarka - Astarka 
Average speed of sun 

Verses 38 - 40 : Kalamsa of the yogatara of 
a naksatra is expressed in kala, multiplied by 1800 
and divided by rising time of its sayana rasi (for 
rising) and by rising time of (sayana rasi + 6 rasis) 
for setting. Result will be ksetramsa in kranti vrtta. 
This will be added to drkkarma dhruva of naksatra 
for rising or substracted for setting. This will be 
udaya or asta dhruva of yogatara. When sun's 
dhruva (polar longitude) is equal to udaya or asta 
dhruva of the yogatara, it will helically rise or set. 

Notes : It has been explained is previous note. 
Udaya dhruva is udayarka, asta dhruva is astarka. 

Verses 41-44 : Extreme cases - Manj naksatras 
in north rise again in east before they set in west. 
Hence their setting is not necessary. Their setting 
has been discussed only to know the setting time 
in west. Udaya and asta of many naksatras like 
Kratu should be calculated. Agastya and yama are 
in far south, hence they remain set for long. 

Day length of any graha or star can be known 
from its carajya calculated from kranti (and local 
aksarhsa). Hence, their daily rising and setting 
times can be known. Still, detailed methods will 
be explained here for their times of udaya and 
asta. 

The discussion so far has been done according 
to the views of earlier astronomers. Now I describe 
more accurate methods thought by me. 



698 Siddhdnta Darpana 

Notes (1) Permanent rising and setling of stars 
has been explained earlier. If kranti of the star is 
more than colatitude of the place, the star will 
never set for north kranti. For south kranti, greater 
than colatitude of the place, it will never rise. 

Equivalent condition is that, astarka 
> udayarka; i.e. star will rise again before it sets 
in west, explained in theorem (1) of previous note 
(3) after verse 37. 

(2) Rising of agastya (canopus) has been 
discussed extensively. According to Aryabhata I, 
Varahamihira and Sumati, agastya rises heliacally 
when 

sun's longitude = 120° + <J> 

and sets heliacally when 

sun's longitude = 180° -(120° + O ) = 60° -O 

where O is the latitude of the place. 

According to Vatesvara; agastya rises heliacally 
when 

sun's longitude = 98° + 42 P/5 degrees 

and sets heliacally when it is 76° - 42P/5 
degrees. 

where P is the equinoctical mid day shadow 
in angulas. 

Manjula gives the formulas as 97° + 8P and 
77-8P. 

Bhaskara II and Ganesa daivajna give 
98°+8P and 78° - 8P 

The above rules have been derived by 
substitution from the following formula 



Rising Setting of Planets, Stars 699 

Udayarka = star's polar longitude + aksa 
drkkarma + kalamsa. 

Astarka = star's polar longitude - aksa 
drkkarma - kalamsa. 

* 

Verses 45-50 - Sphuta kalamsa - 

Planets and stars rise on horizon, when sun 
is still below horizon. Even in such situation they 
are invisible because sun's light reaches on horizon 
(twilight) due to reflection from atmosphere. 

Natarhsa of sun from drk - mandala (when it 
is start of twilight) is multiplied by trijya and 
divided by lambajya. Result is again multiplied by 
trijya and divided by dyujya. Result will be sphuta 
kalamsa of stars from sun. 

This means that, there is big difference 
between drk-mandala arhsa and kalamsa. On 
equator also, it is equal to difference between 
dyujya and trijya. At other places it is still more. 

For example at a place of 66° north aksamsa, 
in mesa month (when sun is in mesa rasi) guru 
and sukra in mlna rasi rise alongwith sun. Both 
these planets are seen only when away from sun. 
Hence, it is not necessary to calculate these kalamsa 
difference in rising times. From the kalamsa written 
for these planets, ksetramsa is more, though 
kalamsa is below the visibility limit. Hence, they 
are seen. 

Notes : Due to reflection from atmosphere, 
twilight starts when sun is still 18° below horizon. 
In India, it is assumed 15° below horizon, as it is 
north of equator. This is called usa in morning 



700 Siddhanta Darpana 

and sandhya is evening. Sandhya is used for both 
twilight periods. 

Then sun rises when it is still about 35' below 
horizon due to refraction of rays in atmosphere. 
Hence twilight period extends from 18° below 
horizon to 35' below horizon position of sun. 

Thus the natamsa of sun below horizon (18°) 
or natamsa of 108° from meridian is the time when 
sun light starts. Thus, it is increase in carajya which 
is equal to increase in half day length. 

Like carajya, the natamsa jya is divided by 
cos <p = lamhajya/trijya to find rising difference on 
diurnal circle. It is divided by cos d = R cos d /R 

= dyujya/trijya to get the degrees on equator 
whose kala is equal to asu time. Hence the formula. 

Here, dyujya on equator means kotijya of 
natamsa, instead of kotijya of kranti. For 66° north 
aksamsa, the difference is Sin 18° /cos 66° = Sin 
30° approx. Hence guru and sukra rise with 30° 
or 1 rasi difference. 

Verses 50-58 - Sphuta dhruva of udayasta of 
graha - From the udaya and asta kendramsa, we 
find the udaya and asta kala of planets. For that 
time, mandaphala of sun is calculated. This 
mandaphala is substracted from fourth sighra 
kendra of guru, mahgala, sani at the time of udaya 
or asta, or added to it in same manner, it is 
substracted or added to sun. In budha or sukra, 
this correction will be in reverse order. 

If graha is less then the true sayana sun at 
that time, half the kalamsa of graha is substracted 
from sun. If graha is more than sayana sun, then 
half kalamsa is added to sayana sun. If sun is in 



I 



Rising Setting of Planets, Stars 701 

east, that will the lagna at that time. When sun 
is in west, 6 rasi is added to sun ± half kalamsa. 
That will be the lagna for setting time. 

3 rasi is substracted from this lagna. Kranti 
of that point of ecliptic (tribhona lagna) is calcu- 
lated. By adding or substracting aksamsa to kranti, 
natamsa and unnatamsa is found (for tribhona 

lagna). 

The natamsa of sukra, guru, budha, sani are 
divided by 4,5,6,7,8 and result is added to 
unnatamsa. Jya of the resulting angle is found. 
Kalamsa of the planets for udaya or asta is 
multiplied by trijya and divided by jya of the 
corrected unnatamsa. 

Result is degrees etc. will be kalamsa of graha 
in ecliptic. This substracted from sun will be dhruva 
for asta or udaya. Half of this asta or udaya dhruva, 
is added or substracted from sayana sun as before 
- That will give corrected udaya lagna or asta lagna. 

Notes : No logic has been given for such a 
long and arbitrary process. Probable justification is 
given below - 

(1) Kalamsa difference from sun is measure 
of decrease in intensity of sun light. Since it 
decreases according to inverse square of distance, 
kalamsa proportinate to birhba diameter (measure 
of intensity) is reduced by half. 

(2) Mandaphala substracted from sighra 
kendra, is distance of planet from sun, on which 
tile brightness of graha depends. 

(3) Natamsa of tribhona lagna is proportional 
to inclination of diurnal circle with vertical. The 
kalamsa will increase in proportion to this obliquity. 



702 Siddhanta Darpana 

It is divided by half the values of kalamsa of graha. 
For bright planet, kalamsa is less, fraction of 
natamsa is more, then corrected unnatamsa and its 
jya will be more, hence sphuta kalamsa will be 
less as it is divided by jya. 

Still the method appears to be based on trial 
and error and probably gave better results. 

Verses 59-68 : Ksetramsa of planets for mid 
Orissa Ksetramsa of planets is being given for mid 
Orissa according to rasi of sayana sun. 



Planet 


Rasi of 


sayana 


sun 


ksetramsa 


Sukra 


1, 12 

2, 11 

3, 10 

4, 9 
5,8 
6-7 






10/26 

9/51 

9/40 

9/20 

9/0 

9/1 



When sun is in west, 6 rasi is deducted from 
it and then ksetramsa is found. Then the degrees 
of sayana sun are multiplied by difference of 
dhruvamsa and added to ksetramsa if increasing, 
or substracted if decreasing. 

Guru 



Budha 



1, 12 


13727' 


2-11 


1372' 


3, 10 


12°11' 


4, 9 


11/28 


5, 9 


1175 


6, 7 


11/0 


1, 12 


16/11 


2, 11 


15/39 


3, 10 


14/32 


4, 9 


13/36 



Rising Setting of Planets, Stars 



703 



Sard 



Mangala 



5, 8 

6, 7 

1, 12 

2, 11 

3, 10 

4, 9 

5, 8 

6, 7 

1, 12 

2, 11 

3, 10 

4, 9 

5, 8 

6, 7 



13/7 

13/0 and 13/1 

18/56 

18/26 

16/53 

15/44 

15/8 

15/0, 15/1 

21/42 

20/53 

19/14 

17/52 

17/10 

17/0, 17/1 



In fifth rasi, astodaya dhruva will be equal to 
madhyama ksetramsa. Difference of 6th and 7th 
rasi has been written as 1' kala. 

Verse 69 : For other places also, from 
unnatajya, ksetramsa between sun and the planet 
for udaya or asta can be found out. 

Verses 70-76 - Sphuta udayasta time. 

When difference of sun and graha is equal to 
dhruvamsa, then that will be the sphuta time. It 
will be made more correct by successive approxima- 
tion. 

From sara of graha or naksatra, both ayana 
and aksa drkkrama correction are done. Both; 
corrections are added together or difference is taken 
according to sign. Resultant correction (positive or 
negative) in vikala is divided by difference of sun 
gati and graha gati in kala. When budha or sukra 
are vakri, it will be divided by sum of gati kala. 



Siddhanta Darpana 

Result in danda etc. is added to rising and 
setting time, if rising is in east and setting in west 
- and when drkkarma was positive. If rising is in 
west and setting in east, then it is substracted from 
rising or setting times. 

For negative drkkarma, reverse in done. The 
correction time is substracted from rising time in 
east or setting time in west. It is added to rising 
time in west or setting time in east. 

Then we get more correct time for udaya or 
asta. This correction is due to motion of planet at 
the cakra or cakradha. 

When sphuta sun and sphuta graha are in 
same rasi then it is cakra time for all tara graha, 
for budha and sukra it can be cakrardha also. 

Note - Ayana and aksa drkkarma have already 
been explained in conjunction of planets. Positive 
' drkkarma means rising of planet is later and setting 
time is earlier, i.e. difference with sun i» reduced, 
(rising in east and setting in west). Then their 
difference will again increase to dhruvamsa distance 
after sometime depending on relative speed. 

Verses 77-82 - Astodaya time without 
drkkarma - For Orissa, udayasta degrees for eacn 
pianet has been stated according to sayana sun in 
each rasi. At the time of cakra, the degrees ot 
udayasta are divided by difference of speeds ; ot 
sun and graha. For cakrardha of budha and _sukra, 
division is by sum of gatis. Result will ^e time ui 
days etc. For that period after cakra or cakrardha 
graha will be set, then it will rise. It will be set 
for that period before cakra/cakrardha also. 



u 



&■■■ 



Rising Setting of Planets, Stars 705 

Sayana sun is calculated for the time of udaya 
or asta found approximately. Again, we calculate 
the difference of sun-graha distance and ksetramsa. 
This is divided by gati antara or sum and udayasta 
times are corrected. 

For this corrected udayasta kala, we take the 
average position of graha and sun. Speed of sun 
and graha for that position is the sphuta gati for 
both for purpose of udaya or asta times. 

At any time we calculate the difference of 
sayana and dhruvamsa corrected sun and the 
sphuta graha. That is converted to kala and divided 
by difference or sum of speeds of sun and planet. 
That will give days since udaya or asta or remaining 
days according to rules explained earlier. 

Notes : (1) At cakra and cakrardha planet has 
same position from earth as sun. Then they will 
be set due to closeness of sun. Assuming the speed 
at end of cakra/cakrardha to be average speed up to 
period of udaya, we calculate the time when the 
planets will be separated at distance of ksetramsa, 
%hen they will rise heliacalry. By successive 
approximation by speeds at approximate time of 
udayasta, we calculate more accurate time. 

(2) The days since udayasta or remaining days 
are calculated by calculating as to when sayana 
Sun - sayana graha = ksetramsa. 

(3) 1/2 (sun + graha) at udaya or asta time is 
the mid position of sun and graha. With sufficient 
accuracy, speeds of sun and graha at that position 
can be considered sphuta. 



706 Siddhanta Darpana 

Verses 82-84 : Conclusion - 

Other astronomers have stated the udayasta 
degrees of graha 1° less than values stated here. 
According to them the degrees are - guru 10, budha 
12, sani 14, mangala 16, vakrl sukra 7, margl sukra 
9. 

This is not the real setting or loss of a planet. 
In course of rotation of earth and their own motion, 
they keep coming east, west, up or down. As eyes 
are dazed due to brightness of sun, tara graha 
become lightless like light flies and become in- 
visible. Due to bigger angular diameters, candra, 
guru and sukra are seen in day light also. 

At equator 22° 30' kalamsa before rise or after 
setting of sun, its light starts reaching horizon. At 
other places this kalamsa is multiplied by lambajya 
of the place and divided by trijya. Light of sun 
will go upto that distance, (kalamsa). 

At equator, light of moon is visible 8 kalamsa 
after, setting or before rise. Light of sukra is visible 
1 kalamsa before rise or after setting. Kalamsa at 
other places is obtained by multiplying it with trijya 
and dividng with lambamsa jya. 

Notes : It ha,s already been explained, how 
planets set hetically. Kalamsa of sun here has been 
taken as 22° 30' at equator against 18° taken in 
modern astronomy. Kalamsa of moon and sukra is 
not calculated, as it is ineffective compared to light 
of other stars. 

At other places, kalamsa depends both on 
kranti and aksamsa as calculated earlier. Roughly 
we can assume that, sun rays will reach same 



Rising Setting of Planets, Stars 707 

manamsa at other places also which is equal to 
Icalamsa X trijya / lambajya 

Verses 85-86 - Prayer and end 

I pray thousand times to revered lord Jagan- 
natha, whose brightness is like jewel of Indra (Indra 
nila mani is blue), whose lotus feet are worshipped 
by Vasuki, Ganesha, Siva, moon and sun. 

Thus ends the thirteenth chapter discribing 
rising and setting in Siddhanta Darpana written 
for correct calculation and a text book by Sri 
Candrasekhara of a famous royal family of Orissa. 




Chapter - 14 

LUNAR HORNS 

Candra Smgonnati Varnana 

(Elevation of lunar homs) 

Verse 1 - Scope - For knowledge of persons 
of sharp intellect, it is proposed to describe accurate 
daily rising and setting of moon, elevation of lunar 
horns (candra srnga) and diagrams (parilekha). 

Verses 2-11 : Time after sun set when moon 
sets. 

Rising time and setting time of moon are 
Calculated roughly according to method described 
earlier. At the time of sun set, accurate moon and 
sun are made sayana (ayanarhsa is added). Then 
drkkarma sanskara is done. 

6 raSi is added to sayana and drkkarma 
corrected sun and moon. By difference of their 
lagna (rising times of rasis between them), kalamsa 
is found out. 

Among sun and moon, bhogya (remaining) 
asu of lesser rasi, bhukta (lapsed) asu of bigger 
rasi and rising time, Of other rasis in between in 
asu - all are added and divided by 60. Result wiU 
be sphuta kalamsa difference between sun and 

moon. 

Kalamsa difference divided by 6 gives the 
result in ghati etc. This is multiplied by gati ot 
sun and moon and divided by 60. Result in kala 






lunar Horns 709 

etc is added to sun and moon. Again, we find the 
difference of their rising times. After repeated 
procedures, rising time difference between sun and 
moon will become steady or fixed. 

Here, ayana and aksa drkkarma of moon is 
to be done every time, otherwise sara will be 
different due to change in distance between moon 
and its path. 

When the rising time difference between moon 
and sun becomes constant thus, 3 rasis are 
substracted from moon. For this vitribha lagna, 
nata and sara are found, and its ayana and aksa 
drkkarma are done. 

To make it more accurate, the drkkarma 
correction is made to moon and sun with 6 rasis 
added to them. The rising time difference in asu 
is found. Sphuta gati of moon is divided by 14 
and multiplied by trijya and divided by lambajya 
to get the lambana asu of moon. On substracting 
this from the rising time difference, we get correct 
difference in asu. 

This period after sun set, moon will set. While 
finding instantaneous sun and moon, asu should 
fee considered savana (21,659 part of savana day) 
and while finding moon at any time it will be 
staksatra (i.e. 21600 part of a naksatra day). At sun 
set time, asu will be candra savana (i.e.. 22, 390 




Notes : (1) Second half of fifth and 1st half 
ff sixth verse are quoted from siirya siddhanta, 
Which was considered by many to be an interpola- 
tion. However, here, they are further specified by 
the values of asu to be taken in these 




710 Siddhdnta Darpana 

calculations. This is the method for calculating 
setting of moon in sukla paksa (bright half). 
Though this is not specified anywhere, but next 
verse tells about procedure for krsna paksa. It has 
been clearly specified in surya siddhanta. 

(2) Rough method for rising and setting time 
of moon - This has been stated in chapter 8 - 
candragrahana verses 60-65. That is for purnima 
and can be used for 8th of sukla paksa to 7th of 
krsna paksa. This ignores sara and doesn't do 
drkkarma sanskara. 

Rising time - At sun set time sayana sun and 
moon are calculated. 

Rising time of moon after sunrise 

= Rising time of remaining part of rasis of 
sayana sun + for lapsed part of sayana moon + 
for rasis in between sun and moon + 56 asu as 
lambana correction for moon = A asu 

= 360 ghatl 

Setting time of moon : Moon set time after 
sunset = rising time of remaining rasis (of sun at 
sun rise time + 6) + for lapsed part of rasi of 
(sayana moon at sun rise + 6 rasi) - 56 asu = A 

asu =^ghati. 

Sara correction - Sara kala X palabha / 12 is 
added to rising time if sara is south and substracted 
if sara is north. Reverse correction is done for 
moon set time. 






711 



f'-l', 

*'■■-' 



t 



M-, ' 



f 
I' ' 



lunar Horns 

Around purnima, moon rise is around the 
time of sunset, hence the position of sun and moon 
at sunset time are taken for better approximation. 

Rising time of moon - rising time of sun in 
east = rising periods of ecliptic between rasi of 
sayana sun to sayana moon. 

Due to larhbana, moon will appear lower 
when seen at horizon and rise 56 asus later or set 
56 asus earlier, the time needed by moon to cover 
earth's radius in its orbit. 

When moon is setting (moon + 6 rasi) is rising 
in east. Similarly at sunset time sun + 6 rasi is 
rising. Hence moon set - sun set. 

= rise of (moon + 6 rasi) - rise of (sun + 6 
rasi) 

= rising time between (moon + 6 rasi) to (sun 
+ 6 rasi) 

For equator, rising time of a rasi and 6 rasi 
away from it is same. 

If we use the time of setting of rasis instead 
of rising, addition of 6 rasis is not needed. 

Sign Time of setting in asus 

'at the equator at the local place 

1. Mesa 1675 

2. Vrsa 1796 

3. Mithuna 1929 

4. Karika 1929 

5. Simha • 1796 
6.Kanya 1675 

Here a, b, c are the rising time differences 
for mesa, vrsa and mithuna. 



1675 +. a 
1796 + b 
1929 + c 
1929 - c 
1796 - b 
1675 - a 



Sign 

12 Mlna 
11 Kumbha 
10 Makara 
9 Dhanu 
8 Vrscika 
7 Tula 



712 Siddhanta Darpana 

Due to north sara, effective kranti is increased, 
hence carajya will increase. Component of sara 
parallel to kranti i.e. perp to equator is s cos e 
where e is inclination of moon's orbit with equator. 
Hence, corresponding carajya increase is 

s x palabha 

s tan <I> = — — 

12 

This is substracted from rising time as day 
length increases due to increase in kranti. 

(3) Successive approximation and drkkarma - 

Due to aksa and ayana drkkarma, difference 
between sun and moon is corrected as visible from 
the place. 

Difference in moon set - sun set 

= . rising time diff (sayana moon + 6 rasi) 
(sayana sun + 6 rasi) 

as the rasi at 6 rasi difference is rising when 
sun or moon are setting. 

That will be sphuta time difference in asu. 

Asu Kala 
— — = —rzr = degree 
60 60 6 

360° = 60 danda (naksatra time) 
Hence 1 danda time = 6° Kalamsa 
Speeds of ravi and sun are calculated for 
savana dina, hence savana asu is to be used (1 
day = 21659 asu). For calculation at sunset time, 
we take candra savana dina because moon set to 
moon set time is equal to candra savana dina. 

From speed of sun and moon at the asta time 
of moon, further corrections are done. 



&' 






|j<nar Horns 71 -3 

(4) Lambana correction at setting time is 
sphuta gati of moon divided by 14. This is lambana 
amsa at local aksamsa. To convert it into kala at 
equator or asu, it is divided by cos <p i.e. R cos<£ 
/ R or multiplied by trijya and divided by lambajya 
= R cos <p. This is substracted from setting time. 

Verses 12-13 '- In krsna paksa, sun at sunset 
time is calculated, 6 rasi is added to sayana sun. 
Difference in rising times between (sayana sun + 
6 rasi) and sayana moon at that time is the time 
after sun set when moon wil rise. 

Here also drkkarma is to be done for sun and 
moon both at sun set time. At rising time, lambana 
asu is added. After repeated calculations with sun 
and moon positions at moon rise time we get 
steady value of diff ernce in sun set and moon rise 
in east. 

Notes : In krsna paksa, difference between 
sun and moon is more than 6 rasi. Hence at sun 
set time, moon is below east horizon, Hence, we 
calculate the difference between east horizon 
ecliptic point (i.e. sun + 6 rasi) and moon. 
However, while the position of moon at sunset 
time comes on ecliptic, moon goes further east due 
to its motion, hence real rising will be later. This 
difference is corrected by successive approximation. 

Verses 14-18 : Position of moon at desired time 

Now, method is described for calculating 
J position of moon at sun set time or any other time 
as observed from earth's surface. 

By method explained in chapter 6, sphuta 
candra is found at desired time. Its position east 
or west half of sky is found (from lagna etc.) 



714 Siddhanta Darpana 

Lagna for desired time, vitribha lagna and 
vitribha sariku is calculated. Drgjiya for moon in 
east or west half of sky is multiplied by drggati 
(vitribha sanku) and divided by trijya. Result is 
multiplied by first sphuta gati of moon and divided 
by 14 X trijya 3438 (=' 48132). Result in kala etc 
will be added or substracted to moon, if moon is 
east or west from vitribha lagna. Then we get 
lambana corrected moon at desired time. After that, 
we find nati of moon again. Nati and sara are 
added or difference is taken according to same or 
different directions to get sphuta sara. From that, 
we do ay ana and aksa drkkarma. By making 
drkkarma correction, we get the samaprota vrtta 
moon as seen from earth surface. 

Notes : Methods of lambana and sphuta 
candra have already been explained in chapter 9 
on solar eclipse. 

Verses 19-27 : Elevation of lunar horns 

There are two types of elevation of lunar 
horns (srnga). Generally horn means, pointed ends 
of the bright portion of the disc. But some 
authorities consider elevation of horns of black 
portion also. This horn is not seen but it can be 
known from calculations. 

* 

From one rise of moon to its next rising time 
is called savana day of moon. 

From sphuta kranti of moon, its nata kala is 
found by method explained earlier. 

In first half of sukla paksa (1st day to 8th) 
and second half of krsna paksa (9th day to 14th), 
elevation of bright horns is found. For other days 
i.e. 2nd half of sukla and 1st half of krsna paksa, 



Lunar Horns 715 

elevation of dark horns is calculated. Out of bright 
or dark parts, whatever is less than half, its 
elevation is calculated. 

Moon in its vimandala (inclined orbit) is 
lighted by rays from sun in apavrtta (kranti vrtta 
or ecliptic) and is seen in many shapes. 

Even when both the horns of moon are 
equidistant from sun, they appear small or big and 
inclined. 

At a place where midday sun at the end of 
uttara-ayana (i.e in sayana mithuna) is above head 
(i.e. aksamsa of the place is equal to maximum 
kranti - karka rekha place), the sun at beginning 
of sayana mesa will be in sama mandala at sunset 
time (i.e. in east west circle). At that place moon 
with zero sara will move exactly in east direction. 

This shows that according to position of kranti 
vrtta, position, speed and horns of moon are 
decided. That also changes due to change of sara. 

As in candragrahana, in finding elevation of 
horns also, ayana and aksa valana are calculated. 

Verses 28-29 - Sara valana 

We consider the right angled triangle whose 
sides are 

(1) Perpendicular side is the jya of difference 
between moon and sun. 

(2) Base is the bhuja of sarajya (or sara with 
direction) 

(3) Square root of sum of these squares is 
karna - i.e. linear distance between sun and moon. 



726 Siddhanta Darpana 

Sara is multiplied by trijya and divided by 
karna. Arc of the result in kala is divided by 60 
to know the valanamsa of sara (i.e. angular 
deflection). 

Notes : Here we form the right angled triangle 
for deflection from ecliptic only. For sun it is zero. 
In surya siddhanta, it is calculated for defleciton 
from equator. For that we take the difference of 
kranti of sun and moon = p 

Then, base 
p x chaya karna of moon ± 12 aksajya 

lambajya 

Perpendicular = Sanku of moon i.e. kotijya 
of natamsa 

Then, Karna =Vbase 2 + perp 2 

Here karna has been termed as madhyahna 
candra prabha karna i.e. straight distance (like a 
light ray) of mid day moon. This has been confused 
with mid day of sun. Ranganatha in his gudhartha 
prakasika tika on Surya siddhanta, interprated it 
as mid point of civil day between sun rise to next 
sunrise i.e. sunset time. Accordingly, he derived 
the formula. This was followed by Burgess who 
wrote the commentary in 1860 at Chicago U.S.A. 
after he got Ranganatha Tika in Maharastra in 1835. 
Svami Vijnanananda followed it in his Bangala 
commentary in 1909 and Sri Mahavira Pd Shrivas- 
tava in his Vijnana Bhasya in 1940. 

When moon is at meridian or its midday, sun 
is at horizon or above it i.e. within ± 90° of moon, 
as the bright portion is less than half for a horn 



Lunar Horns 717 

to follow. Hence it will be almost correct for other 
positions also. 

Proof : 

SZ is a quarter of the yamyottara vrtta. Let 




Figure 1 

C be its centre and Z zenith. Let EC be nadi 
mandala. ZC is produced to D, so that CD 
represents a sariku of 12 angulas. 

When sun is at E, DF is equinoctical shadow 
of CD or palabha. When sun is at A, DG called 
bhuja is shadow and GF agra. When sun is at B, 
DK called bhuja is shadow and FK agra. 

Thus Bhuja = Palabha ± Agra; or ± Agra, 
when sun is on horizon. 

Similarly bhuja of moon = Palabha ± moon's 
agra in the sphere whose radius is candra chaya 
karna i.e. hypotenus of right angled triangle whose 
one side is sanku of 12 angula and other is shadow 
caused by moon. 

Thus in this sphere, sun's agra 
Sun agra X candra chaya karna 



Moon's agra = 




Candra agra x Candra chaya karna 

Tnjya 



718 Siddhanta Darpana 

tw aOT3 Kranti jya x Trijya 

But agra = . _ 

Lambajya 

Hence difference between sun's and moon's bhuja 

= Palabha ± (Candra kranti jya ± Sun kranti jya) 

Candra chaya karna 

Lambajya 

Palabha 12 

, , . _ = ~ : — rr , hence we get the 

aksajya Lambajya ° 

formula for base. 

From this base and sanku of moon's height, 
we get the karna which is direction from sun to 
moon in meridian circle, i.e. projection of sun moon 
line in this circle. 

(2) Due to the confusion about this interpreta- 
tion and apporximate formula, siddhanta darpana 
has given more direct and accurate formula which 
can be used for any position of sun and moon. 




Figure 2 • 

It is known that sun is always on ecliptic, but 
position of moon at M on ecliptic is perpendicular 
foot on ecliptic. Thus MM' is perpendicular on 
plane of ecliptic, i.e. on line SM' of this plane also. 

Arc SM' is difference in moon and sun on 
ecliptic i.e. their rasi difference. Line SM' is the 
jya of that difference. MM' is sara of moon i.e. 
sara jya (arc MM' is the sara). 

Hence SM = ^$M' 2 + MM' 2 



lunar Horns 719 

gives prabha karna of moon at any time. 

(3) When we know the karna, at this distance 

sara 
gara will make an angle = radian 

sara x trijya kala . 

= - in kala 

karna 

■ 

We are following the scale of 1 angula = 1' 
on khagola circle in diagram. Hence, kala is 
converted to degree or angula by dividing it with 
60. 

Verses 30-43 - Diagram of lunar horns 

We draw a khagola circle for same radius 
(57° 18') as in diagram of eclipse. Directions are 
marked. 

Here also, moon is shown as a circle of radius 
6 angula (i.e. 12 angula diameter). When moon is 
in east kapala, sphuta valana is given in eastern 
point and if moon is in west half of sky, valana 
is given near west point in its direction (north or 
south). 

Valana of sara is given in opposite direction 
from valana given earlier in both kapala s (east or 
west half of sky). 

End point of sara valana is assumed to be 
sun and from that, a line is drawn upto centre of 
moon and extended. The point where it cuts the 
circumference of moon will be the border point 
between bright and dark portions of moon due to 
sun. 

Difference of moon and sun in kala is divided 
by 900 to get the angula width of bright portion. 
From centre of moon, on the sun line (the end 



720 Siddhanta Darpana 

point of sara is sun), we give two points on kha- 
vrtta at 90° distance from sun on both sides. From 
these points also two lines are drawn to the centre 
of moon. 

These two points cut moon on ends of a 
diameter. On sun line, from circumference, a point 
at distance of width of bright portion is given. 

To draw a circle through these points, we 
draw arcs with 5 angula radius from each of three 
points. They form two fish figures, whose head - 
tail lines cut at the centre of circle through these 
points. From the arc through the three points, the 
portion towards sun will be the bright portion of 
moon. 

In sukla paksa, if moon is in western sky, or 
in krsna paksa moon in east sky, the bright side 
will be towards sun point. For sukla paksa moon 
in east, krsna paksa moon in west, the bright side 
of moon will be on opposite side of sun point. 

In sukla paksa, less than half bright moon 
will be shown by putting the diagram on west side 
wall. Horns will be bent towards north or south. 

In krsna paksa, this will be shown on eastern 
wall. For more than half portion of moon bright, 
it will show elevation of dark horns. 

Notes : (1) Valana of moon depends on aksa 
and ayana valana and due to its sara from ecliptic. 
Hence both are marked. 

Since moon circle is of 12 angula diameter, 
complete diameter 12 angula will be bright when 
moon - Sun = 180° = 10,800 kala 



lunar Horns 



721 



Hence 1 kala difference = 



12 
10,800 



angula bright 



part 



900 



bright part. 



This assumes that moon's speed is constant, 
which makes little error. But another assumption 
is that bright part is proportional to angular 
difference between sun and moon. Actually, it is 
proportional to utkrama jya i.e. R (1-cos ) as 
shown below. 



#■.•■. 




L 1 ' ■ 



Figure 3 - Phase of moon 

C is centre of moon, CO is direction of 
observer, BEAF is the face of moon, perpendicular 
to direction of observer. CS is direction of sun and 
EMF is the face of moon perpendicular to CS 
direction of sun. Thus the portion of moon between 
EMF and EBF is the bright portion seen to observer. 
But the circle EMF is seen obliquely by observer, 
hence it is seen as half ellipse as projected on 
BEAF plane whose major axis is EF and semi minor 
axis is CN. This projected ellipse ENF is the internal 
boundary of bright portion. CM is radius of moon 
hence equal to CB and CN is projection of CM, 
hence 



722 Siddhanta Darpana 

CN = CM cos MCN = CB cos SCC 
Because iLMCN is the angle between planes 
which are prependicular to the directions of 
observer and sun. Hence bright portion NB 
= CB - CN = CB - CB cos SCC 
= CB (1 - cos SCC) = CB. Utkramajya SCC 
Z.SCC is roughly the angle between direc- 
tions of sun and moon. If they are considered in 
ecliptic it is difference between longitudes. More 
accurately, it can be found from triangle OCS 
whose sides OC, OS and CS are known. 

(3) In first half of sukla paksa, when sun is 
setting, moon will be in west half of sky as it is 
less than 90° ahead of sun. Hence, the diagram 
will be shown on west wall with direction of sun 
downwards. In later half of sukla paksa, moon will 
be in east half of sky, hence its dark horns will 
be shown in east sky because more than half part 
in bright. 

Verses 44-61 - Modern method of showing 
lunar horns. Thus the method for finding lunar 
horns has been described according to old 
siddhanta texts. Now, I describe accurately, ob- 
served bright part of moon according to my 
experience and logic. 

Jya of difference of moon and sun ra& etc is 
multiplied by yojana karna of moon and divided 
by yojana karna of sun. Result in kala etc is added 
to moon of sulda paksa and substracted from moon 
of krsna paksa. Ihat will be sphuta moon. 

From this sphuta moon, rasi of sun is again 
substracted and utkxarna jya is found. That is 



lunar Horns 723 

divided by 573. Result in angula etc. will be 
measure of bright portion or dark portion which 
ever is less than half. 

When less than half of moon is bright, this 
resultant angula will be marked as bright portion. 
If more than half is bright, then bright angula 
measure is (6 angula - the result). 

As before, from the end point of sara valana 
in the direction of sun, three points on bright dark 
boundary are found. Through fish lines, we find 
ihe centre and draw a circle through these points. 

Bright portion of moon less than 1-1/2 angula 
Yt.e. l/8th of moon's diameter) is not seen, because 
|he end portions of horn are very thin. Increase 
|n phase of moon, or its decrease should be shown 
Jo people through diagrams. On 4th day of bright 
half (sukla paksa), at the sunset time, moon circle 
is drawn in north direction on earth's surface. 4 
diameters are drawn through directions points and 
angle points. All the diameters bisect each other 
at the centre. West from moon at a distance, sun 
is shown. Due to this sun, west half of moon will 
be bright. To see the bright portion, earth point 
is given at a distance of 5 hands (5 X 24 angulas) 
from moon's centre in agni kona (south east 
direction). North of this earth point will indicate 
zenith of sky. Though half the moon is always 
lighted to sun, the portion seen from earth is much 
ptess than half due to angle between white circle 
and visible circle. From southeast direction, we see 
the diameter through nairtya (south west) and isana 
points (north east). Of the bright portion touching 
|lP*e north south line, lower half portion will be 
from earth. 



;a - 



M 



724 Siddhdnta Darpana 

The line from earth centre to south point of 
moon touches west point and cuts the north east 
- south west line. From this point in direction of 
south west, bright portion will be seen. Rest part 
upto north east point will be dark. 

When difference of moon and sun is 45°, 
ancient texts, have assumed 3 angula bright 
portion. But in this calculation only (1/45) angula 
is actually seen. Hence, scholars calculate the bright 
portion of moon from utkrama jya only, because 
in a sphere, any object will be seen in line of sight 
(in perpendicular plane only). 

Notes : (1) Use of utkrama jya - The formula 
as proved in previous section is through utkrama 
jya as shown, Logically we can infer it because we 
see the sphere from curved side, not from side of 
centre. Hence the distance of plane surface will be 
proportional to utkrama jya, from centre it is 
proportional to kotijya. 

(2) Proof of the formula- 

As in figure 3, we need 
to know the angle C C S as 
seen from moon between 
directions of sun and direction 
from observer. 

L SCC = L CSO + 
L COS - (1) 

L COS is the angle be- *' 
tween directions (rasi) of sun Figure 4 . angle from moon 
and moon). between sun and earth 

In sukla paksa (moon - sun) is less than 180°, 
hence it is smaller angle COS itself. In krsna paksa 
it (moon - sun) is more than 180°, hence we 




lunar Horns 



725 



calculate the outer angle (360° - ZCOS) Then 
ZSCC = iiCOS - ZCSO - - - (la) 

Now in ACOS, by sine' rule 

PC OS OS 

Sin CSO " Sin OCS ~ Sin SCC 

as Sin SCC = Sin (180°-OCS) = Sin OCS 

or, Sin CSO = OC/OS Sin SCC 

^ L CSO is very small, because OC is very 
|mall compared to OS. Hence Sin CSO = L CSO 
sin L SCC = L SOC approximately. 

Then L CSO = OC/OS (L SOC) (2) 

Here, OC = candra karna, OS = surya karna 

if- Putting the value of L CSO in (1) or (la), 
we get the SCC whose utkrana jya is to be found. 

(3) Ahgula value of bright part. 

For angle of 90°, utkrama jya is 3438 kala and 
bimba is 6 ahgula bright 

Hence for utkrama jya 1 kala, 

6 1 




K: 






bright portion is 
(4) Diagram 



3438 573 



ahgula 



hi 

$:■ ■■ ■ 



ir- 






tr 



*■■■ 



m 




ToEMti 

Figure 5 - Bright portion smo from earth 



726 



Siddhanta Darpana 



NWS is the face to wards sun and is bright. 
Face SW-S, E - NE is towards earth. Hence west 
of point S only, bright portion of moon is seen. 
WS line is hence the boundary of bright portion, 
it cuts SW-NE line on K. Hence from K to SW, is 
bright portion and remaining part from K to NE 
is dark portion. 

(5) Modern method - The great circle from 
sun's centre to moon's centre is perpendicular to 
line joining lunar horns. The great circle from 
zenith to centre of moon is at angle from sun 
moon great circle, which is the angle of lunar horns 
with horizon. This angle can be known from 
spherical trigonometry, as discussed in triprasna- 
dhikara. 




Figure 6 (a) 




Figure 6 (b) 



IN figure 6 a, 

NZS = yamyottara, Z = Zenith (Khasvastika) 

O = observer, NOS = north south line 

NWS = western horizon 

M = Moon in western sky 

R = position of setting sun 

ZM = natamsa of moon 

MR = Distance between sun and moon 



lunar Horns 727 

L RZM = Difference between directions of 
moon and sun (digamsa) from zenith. 

Natakala can be known from visuvamsa (rising 
times) of sun and moon and their kranti. 

cos (nata kala) 

cos (natamsa) - Sin (akSamsa) x Sin (kranti) 
cos (aksamsa) x cos (kranti) 

This equation will give the natamsa 

Then digamsa will be known from the 
following equation 

cos (digamsa) 

cos (dhruvantara) - cos (natamsa) x Sin (akSamsa) 
Sin (natamsa) x cos (aksam§a) 

These equation have been derived for calcula- 
tion of natamsa and calculation of karna vrttagra 
in Triprasnadhikara verses 71 notes (3) and verse 
44 (notes). 

Thus in spherical triangle ZMR, we know ZM, 
ZR, MR, and L MZR. ZR = 90°. Hence we can 
know L ZMR and the elevation of lunar horns. 

In figure 6(b), M = centre of moon 

OM = vertical circle of moon centre 
(drk=mandala) 

RM = Direciton of sun from moon 

ANKBC = Bright portion of moon 

L OMR -L AMN = angle of elevation of 
lunar horns 

In figure 6 (a), from spherical trigometry 

cos MR = cos ZR cos ZM + Sin ZR Sin ZM 
Cos L RZM 

After finding MR from this equation. 



728 Siddhanta Darpana 

Cos ZR - cos ZM x cos MR 
cos L ZMR = SinZMxSinMR 

180° - L ZMR is the angle of elevation of 
horns, because it is equal to L PMR. If sun is 
north from moon, then north horn will be upper 
and if south, then south horn will be upper. If 
digamsa of both sun and moon are same then 
horns will level. After knowing this, diagram of 
horns should be drawn as per figure 6(b). 

Verses 62-63 : Horns of budha and sukra also 
are visible through telescope. 

In India, north horn is mostly seen higher in 
both west and east kapala. Very rarely, south horn 
is seen higher. 

Notes : For aksamsa more than 28-1/2° north, 
both sun and moon will be always in south. As 
we see from north, northern portion of bright horn 
will look bigger. 

Verses 64-67 : Reasons for new methods 
— Earlier astronomers used to find difference of 
kranti's of sun and moon through a sanku of 12 
angula and from that, elevation of horns was 
found. Since this method doesn't give results as 
observed, I am rejecting it. When sun is prepen- 
dicular to equator, half disc of moon in sayana 
makara beginning is seen cut by meridian line at 
zenith. Hence, half disc is seen bright. 

Hence utkrama jya of (moon-sun) is multiplied 
by 1st sphuta gati of moon and divided by (173452). 
Result in kala is added or substracted from half 
disc of moon to find the bright portion width. This 
is added to half diameter when bright portion is 
more than half, otherwise substracted from it. This 



&*":.■ 



729 
lunar Horns 

w fll be correct measure of bimba in both east and 

west sky. 

When moon is at 11° from sun, its light is 
more than budha bimba of diameter 17 vikala and 
less than bright bimba of guru. Hence, it is not 
proper to consider heliacal rising and setting of 
moon at 11° kalamsa difference. The author 
considers it to be between 11 P and 12°. 

At the end of 1st day of bright half, 108th 
part of moon's disc is seen, even though it is very 
thin. On 4th day, its 1/6 part will be seen bright. 
hi the end of 5th day, 1/4 parts will be bright. At 
the end of tenth day 3/4 part will be bright. On 
purnima, complete disc will be bright. On 8th day 
end* half disc and on 11th day end 5/6 parts will 
I appear bright. 

^ Notes : (1) Brightness of moon has been 

calculated according to value of utkramajya for the 
angle between sun and moon. 

(2) Madhya bimba kala x madhya karna of 
candra = spasta bimba kala x spasta karna 
Hence bright portion in kala 

spasta bimba kala x utkramajya 

2 X trijya (= madhya karna) 
madhya bimba kala x utkrama jya 
* 2 X spasta karna 
= madhya bimba kala X utkramajya 

sphuta gati 

2 X trijya X madhyagati 






&' 



fe 



b ■ 
>■; ■ 



730 Siddhdhta Darpana 

- utkramajya X sphuta gati 
madhya bimba kala 

2 X trijya X madhya gati 
= utkramajya X sphuta gati 

444 x trijya 



48705 X X 2 x trijya x 790/35 
utkramajya x sphuta gati 
= ~~ 173452 "as given 

Verses 68-69 - Prayer and end 

On sea beach, Lord Jagannatha protects 
people from anger of yama with his sudarsana 
cakra, and destroys aU diseases borne out of 
desires. May he end all our illnesses due to 
passions. 

Thus ends the fourteenth chapter describing 
elevation of lunar horns in Siddhanta Darpana, 
written for consonance in calculation and observa- 
tion and education of students, by Sri 
Candrasekhara born in famous royal family of 
Orissa. 




£v 






Chapter - 15 

MAHAPATA VARNANA 

Verse 1 - Scope - I am describing mahapata 
!# told in scriptures, which destroys the good deeds 
(karma) earned in pilgrimage, sacred thread wear- 
marriage etc, in whose discussion mathe- 
Hans are also confused, and on whose 








C]- 



occasion, results 01 cnaruy, jap* anu uaui i/™^«- 
auspicious as in an eclipse. 
Notes : Mahapata is a fictious conjunction of 
Sun and moon and is as good or bad as an eclipse, 
lestroys results of good deeds which accrue due 
marriage etc as described in scriptures. But if 
|§pod works like charity are done during mahapata, 
pfhey are as fruitful as in eclipse. This is a difficult 
lopfc, as the conjunction is observed only mathe- 
matically not as a real phenomenon. 

Verses 2-8 : Two mahapatas - 

Patas are of two types - Vaidhrti and vyati- 
pata When their (sun moon) krantis are equal, then 
these patas occur. Out of gola and ayana, if ayana 

ie, then pata is vaidhrti and if gola is same 

then it is vyatipata. 

When moon and sun are in same diurnal 
they have gola sandhi. When both are in 
ce of parama kranti, they have ayana sandhi. 

When moon and sun are in one gola but 
different ayanas and their krantis are equal then 
it is vyatipata yoga. 



■'.'<-■ 







732 



Siddhdnta Darpana 



When moon and sun are in different gola but 
same ayana and their kranti are equal, it is vaidhrti 
yoga. When kranti is same, their aspects are added 
(i.e. they are at same angle with equator plane). 

(In Surya siddhanta) when moon and sun 
both have same kranti, due to combination of their 
rays at same angles there is flow of fire which is 
destructive for living beings. 

Atipata yoga is always bad and destructive. 
Other names of this yoga are vyatipata and 
vaidhrti. 

Each pata has dark colour, very ferocious body 
and red eyes. Both are valiant and occur every 
month. Pata from spasta position (of moon and 
sun) is more destructive than pata from mean 
position. (Quotation ends) 

Notes (1) Two yogas are named vaidhrti and 
vyatipata, but these have no relation at present 
with the two mahapatas. However, these can be 
calculated from sum of longitudes of sun and moon 
and in that way they are related to yoga cycle. 

dinemal circle 
of sun 



diurnal 
circle 





Equator 



diurnal circle 
of moon 



Figure 1 - (a) Vyatipata Figure 1 - (b) Vaidhrti 

Vyatipata is 10th yoga and vaidhrti is the last. 



tfghapata Varnana 733 

Figure 1 - Mahapata 

Figure 1 (a) shows vyatipata when, moon and 
sun have common diurnal circle i.e. same kranti 
tyxt at the other end of orbit. 

Figure 1 (b) shows, vaidhrti yoga, in which 
the kranti of moon and sun are equal and opposite, 
|.e. diurnal circles of moon sun are at equal 
distances from equator, but in opposite direction. 
Sun and moon are in same side of sphere i.e. in 
same gola. 

In these figures Y is 0° sayana mesa and Q 
Us pata of moon i.e. rahu. £ is inclination between 
§|quator and ecliptic and § between ecliptic and 
Inoon's orbit. 

If latitude of moon's orbit is neglected (it is 
|ess than 5° always), both moon and sun are on 
eelitpic. If their longitudes S 2 of sun and M 2 for 
pinoon from sayana 0° are taken, then for equality 
I ©f kranti. 

I Sin Sl = Sin Ml 

I = Sin (180° -M L ) 

I or S L = 180° -Ml or S L +M L = 180° 

?■■"■ 

I When they are numerically equal but in 

| opposite direction, then Sin S L = - Sin M L = Sin 

|<360°-Ml) 

I . Hence S L + M L = 360°. 

(2) In vaidhrti yoga, Sl + Ml - 2 ayanamsa = 



360 



180 



For Vyatipata yoga Sl + Ml - 2 ayanamsa = 



Thus vaidhrti yoga coincided with vaidhrti 
f ***ahapata when ayanamsa was 0°. But vyatipata 



734 Siddhdnta Darpana 

yoga is only 10th yoga starting at 120° and it will 
not tally with the mahapata. 

Verses 9-15 : Calculation of yoga - 

When sum of sayana sun and sayana moon 
is 12 rasi then vaidhrti yoga is near. 

Similarly, if sum of sayana sun and sayana 
moon is 6 rasi or 18 rasi, then vyatipata yoga is 
imminent. 

When sun and moon are in different quad- 
rants of ecliptic, then only pata can happen. Both 
vyatipata and vaidhrti yogas occur once each ( 
month. In some months vyatipata occurs twice, 
sometimes it doesn't occur in a month. 

Pata are possible when viskambhaka etc yogas 
occur. For that, we multiply ayanamsa by 2 and 
added to minutes (kala) of a circle (21,600) or half 
circle (180° = 10,800') if substracted earlier, and 
substracted if added earlier. When result is more 
than (21,600), cakra (21,600) is substracted/ By 
dividing it with (800), result will be past no. of 
yogas from viskumbha etc. Adding 1 to quotient 
it will give the number of current yoga. Remainder 
multiplied by 60 and divided by 800 gives the part 
of current yoga lapsed. 

If moon has no sara, then it is also the time 
of pata. When moon has sara, pata will be slightly 
before or after this time. Hence we should roughly 
calculate pata, first according to madhyama kranti 
(i.e. Kranti of ecliptic point of moon without sara). 

Notes : If has been explained earlier that patas 
will occur when sum of sayana moon and sun is 
6 rasi (for vyatipata) or 12 rasi for vaidhrti. 



Mahapata Varnana 735 

For, if Sl amd Ml are sayana longitudes of 
$un and moon, when their krantis are equal, for 
vyaupata 

Sin S L = Sin M L = Sin (180°-M L ) 

or S L = 180°-M L or S l + M L = 180° (1) 

When kranti is equal and opposite for vaidhrti 
Sin S L = - Sin M L = Sin (360°-M L ) 
or Sl+Ml = 360° (2) 

While pata is calculated with sayana sun and 
moon, assuming madhyama kranti without sara, 
yoga is calculated for nirayana moon and sun. 

Hence, if ayanamsa is A and nirayana moon 
:and sun are S and M, then 

>-'■' (S + M) Kala 

y°8 a= SOOKala " * " ' (3) 

because each yoga extends for 800 kala of sum 
of sun and moon position. 

S L = A+S, Ml = A+M 

Hence for patas 

S L + M L = 6 rasi or 12 rasi (180° or 360°) 
or (S+A) + (M+A) = 6 or 12 rasi 

or S+M = (6, or 12 rasi) - 2 A 

Putting this value of S+M in kala in (3), we 
get the yoga number as stated in text. 

Formulas (1) and (2) give kranti depending 
only on ecliptic. Since sara is very small it will be 
approximate time of pata also. As ayanamsa 
remains almost constant, the yogas for occurrance 
of pata are fixed for some years. We can thus 
know the approximate time of pata by the current 
yoga. After knowing sthula pata, we get it corrected 



7 36 SiddMnta Darpana 

for sara of moon to know when sphuta kranti is 
equal. 

(2) Since yoga is sum of sun and moon, it 
changes with sum of speeds i.e. (790/35+59/8) = 
849/43 average speed. At this rate, rotation takes 

21,600 Kala = ^ approximately 

849/43 kala/day 

In a lunar month of 29.5 days it will definitely 
complete one cycle, hence both the patas will occur 
once at least. Due to extra length of lunar month, 
sometimes, one pata may occur twice. If true kranti 
of moon is more than 23-1/2°, a pata may not 
occur. 

Verses 16-20 : Sthula pata for present ayanamsa. 
At present (1869 AD - writing of book), 
ayanamsa is 22°. (It can be almost same in 1996 
also with only 1° difference). Hence in sukla (24th 
yoga) and vrddhi (11th yoga), vaidhrti 3rd quarter 
and vyatipata 1st quarter often fall. That is their 
madhyama time. 

Hence, we assume the sukla yoga and vrddhi 
yoga as cakra (21,600) and cakrarddha (10,800 kala) 
aproximately. On the day of that yoga, we calculate 
accurate value of sun and moon (at the end ot 
these yoga times). Ayanamsa is added to both sun 
and moon. Then we find the difference of (sayana 
sun+moon) from (10,800) or (21,600) Kala. That wiU 
be divided by sum of' sun and moon gatis and 
multiplied by 60 to get time in danda etc. (It can 
be calculated from proportionate duration of cur- 
rent yoga also) This time is added to time of sukla 
or vrddhi yoga if (sun+moon) was less than that, 
otherwise it will be substracted. 



Mahapata Varnana 737 

After successive approximations, sum of 
say ana sun and say ana moon will be equal to 6 
or 12 rasis at the calculated time. Then we calculate 
the sara of moon. 

These yogas are not visible, hence drkkarma 
or lambana, nati are not needed for moon. Pata is 
calculated from earth's centre only. 

Notes : According to method described after 
verse 15, the yogas at the times of pata have been 
calculated (based on madhya kranti of moon, 
assuming zero sara), for 22° ayanamsa. At present 
also for 23-1/2° ayanamsa, it is almost same. 

Accurate time of madhyama pata is found by 
method of successive approximation. 

Verses 21-33 - Pata from sphuta kranti 

(From Surya siddhanta) - In odd quadrants, 
if sphuta kranti of moon (i.e. kranti of its ecliptic 
point corrected for sara) is more than kranti of 
sun, then pata has already passed. If sphuta kranti 
is less, then pata is yet to come. In even quadrants 
if sphuta kranti of moon is more than kranti of 
sun, then pata is to come, if it is less then pata 
has passed. 

Persons conversant with gola (spherical 
trigonometry) can know the time of sphuta pata 
through their methods. But detailed calculation 
method is explained for common men. 

When sum of rasis of sun and moon (both 
Myana) is exactly squal to cakra or cakrardha kala 
^,600 or 10,800 minutes), then if pata has lapsed, 

60 danda is substracted from that (mean pata) 
me. If pata is yet to occur, then 60 danda is 





in*" 




7 3£ Siddhanta Darpana 

added to that time. For that revised time, we 
calculate sun, moon's pata and sara and difference 
between sphuta kranti of sun and moon. If sign 
of (candra kranti - sun kranti) has changed after 
this revised time then, pata has occured during 
this 60 danda interval. If sign is same, then pata 
is beyond that interval. 

To find the correct time of pata, we find the 

difference of krantis of sun and moon, both at the 

mean pata time and at interval fo 60 danda. If 

they are of different sign, they are added. If 

difference is of same sign their difference is taken. 

This will be the first kranti gati for finding pata. 

First kranti difference in kala is multiplied by 

60 and divided by first kranti gati. Quotient in 

danda etc is added to mean pata time, if pata was 

to come and substracted from it, if pata had already 

passed. 

At first corrected pata time, we again find the 
kranti difference of sun and moon and find the 
second kranti gati kala. Kranti difference of 1st 
corrected pata time is multiplied by the time 
difference and divided by second kranti gati. By 
the result in ghati etc, we again correct the 1st 
corected pata time. Kranti gati is found by 
multiplying the change in kranti difference by 60 
and dividing by the time difference. 

By repeating this process, by successive 
approximation we get the time of mid-pata. Last 
kranti gati wil be the gati of kranti antara at mid 
pata time. 



Mahapata Varnana 739 

B 



V 




+ 




C 

Figure 2 - Mahapata 

Notes : (1) Whether pata has gone or not - 

In figure 2, VBAC is the ecliptic where V is 
vernal equinox, (or say ana mesa 0°) and A is 
autumnal equinox. B and C are solstice points in 
summer and winter at 90° from these. Thus the 
1, 2, 3, 4th quadrants from V are VB, BA, AC and 
CV in the direction of motion shown by arrows. 

When moon is in odd quadrant (VB or AC), 
eg. M in VB, then for vyatipata sun is at Si so 
that VM = ASi and VM+VSi = VM + VA - AS = 
VA = 180°. Similarly for vaidhrti, sun will be at 
S2 in VC where VS2 = VM. Thus sun will be in 
2nd or 4th quadrants i.e. in even quadrants. 

At V and A, kranti (madhya kranti for moon) 
is zero, in VB portion it increases in north direction 
and in AC portion in south direction. Thus the 
kranti increases in VB and AC which are odd 
quadrants and decreases in the even quadrants BA 
I and CV. 

Thus when moon is in odd quadrant and its 
true kranti is more than sun (when madhya kranti 
is equal) then kranti of moon will further increase 
and sun will decrease for even quadrant. Hence 
they will be equal at an earlier time i.e. pata had 



&■ 



'ft/ 



740 Siddhanta Darpana 

already passed. If moon's true kranti is less than 
sun, it will increase and sun's kranti will decrease 
and they will be equal after some time. Hence pata 
will come after some time. 

When moon is in even quadrant, sun will be 
in odd, hence moon kranti will be decreasing and 
sun kranti will be increasing. If moon's kranti is 
more, it will be equal to sun after some time and 
spasta pata will come. If moon's kranti is less, pata 
has already passed. 

This analysis has considered increase of only 
mean kranti of ecliptic point. Sara of moon also 
changes, which will change the true kranti. Hence, 
for correct calculation, moon's sara also has to be 
calculated. 

Suppose moon in first quadrant has 5° north 
sara (maximum value). Then its true kranti at 
madhyama pata will be 5° more than sun. Moon's 
pata decreases at average rate of 5/6.8 degrees 
perday, because its quarter revolution is in 27.3/4 
= 6.8 days. Kranti of sun will decrease and madhya 
kranti of moon will increase at the rate of 23.5/91 
= 1/4° per day approximately. Hence total increase 
in moon's Kranti will be 5/6.8 + 1/4 + 1/4 per day 
= 1.24° per day compared to sun. Thus the true 
pata will be about 4 days before madhyama pata. 
Suppose it is vaidhrti pata. If previous vaidhrti is 
3 days later, then they will be in 25-7 = 18 days 
and in 12 days of the month another pata can 
occur. Thus there will be two vyatipata which 
comes about 12 days after vaidhrti and one has 
already passed between two vaidhrtis. 



Mahapata Varnana 

(2) Calculation of true pata time 



742 




To + 60 



Fig - 3a 



To + 60 




Fig - 3 (b) 



B 



Figure 3 - Calculation of true pata time 

To is the madhyama pata time when mad- 
hyama kranti of sun and moon are equal. But the 
true kranti of sun and moon are unequal due to 
sara of moon. Let AT be the difference of kranti 
of moon and sun at time To (it is negative if kranti 
of moon is less). To make a first aproximation of 
true kranti time, we calculate the position at 60 
ghati difference according to the pata time is earlier 
or later. Then kranti difference is BB' where B' 
indicates time To ± 60 danda. When kranti 
difference has same direction i.e. at O (To) and B' 
(moon-sun kranti) is both positive or negative, the 
true kranti will be equal at time C outside B' i.e. 
outside the interval of T to To ± 60. We assume 
here that kranti difference has same gati (or rate 
Of change), hence it will be zero where line AB 
Guts the line OB' where kranti diff. is zero. This 
is shown in figure 3(a). 

Figure 3(b) shows that the sign of kranti diff. 
;es. Then AB line cuts OB' between the 
Interval at C. 




742 Siddhanta Darpana 

In both the figures we draw a line BA' parallel 
to OB' which cuts AO (or AO extended in fig b) 
at A' Then AA' is change in Kranti diff. in 60 
danda time. Here AA' = AO-BB' = diff of kranti 
diff in fig (a) when kranti difference has same sign. 

AA' = AO + OA' = AO .+ BB' = sum of kranti 
diff. in figure (b) when they are of different signs. 

Thus speed of kranti diff. is AA760 in each 
danda. Hence it will be zero in time T Ti 

* ■ 

AO x 60 J , 
= danda 

AA' 

Here AA' is the gati of kranti antara in 1 day 
or 1st kranti gati. 

Thus we correct the madhya pata time 
according to difference of kranti antara in 1 day. 

By calculating the kranti difference again at 
point Ti we get more accurate value of true pata. 

(3) Surya siddhanta has given another 
method, using difference of moon from rahu. Here 
we have not described the method of calculating 
sara of moon, which is necessary for sphuta kranti. 
Sara depends upon bhuja jya of difference between 
moon and rahu, hence, we take this as difference 
of sphuta kranti in surya siddhanta. 

Verses 34-42 - Sparsa and moksa of pata. 

This was the time, centres of moon and sun 
were having same kranti i.e. mid point of pata. 
When the first points of moon and sun have equal 
kranti, this is called sparsa time as in eclipse and 
When the last point has equal kranti it is moksa 
time. Thus full pata time is from sparsa to moksa. 



Mahapata Varnana 743 

Now, method to find sparsa and moksa time is 
being described. 

Like method of lunar eclipse, we find the 
bimba of moon and sun at mid time of true pata 
and add their semi diameters (manaikyardha),. Sum 
of semi-diameters is multiplied by 60 and divided 
by last kranti gati (i.e. kranti difference gati at mid 
pata time). 

Result will be madhyama sthiti ardha time in 
danda. By adding to pata mid time, we get moksa 
time and by substracting we get the sparsa time. 

At these approximate times of sparsa (or 
moksa), we again find the difference between 
sphuta krantis. If this is less then manaikyardha 
(sum of semidiameters of sun and moon) then 
sparsa time has passed (or moksa time is to come), 
as the kranti difference decreases from sparsa time 
(equal to manaikyardha) to mid time of pata, where 
it is zero. When kranti antara is more than 
manaikyardha, then sparSa is to come (or moksa 
time has passed). 

Kranti antara at sparsa (or moksa) time 
multiplied by 60 and divided by madhya sthiti 
ardha will give 1st kranti antara gati at sparsa (or 
moksa) 

Manaikyardha vikala at sparsa (or moksa) is 
divided by first kranti antara gati at sparsa. It will 
give sthiti ardha for sparsa (or moksa) in danda. 
By substracting them (or adding) to mid pata time, 
we get the time of sparsa (or moksa) - 1st sphuta 
value. 



744 Siddhdnta Darpana 

Now at the first sphuta value of sparsa (or 
moksa), kranti antara kala is multiplied by second 
sthiti ardha time in ghati for sparsa (or moksa). 
We get second kranti antara gati at sparsa (or 
moksa). Again we can get second sphuta value of 
sparsa (or moksa) times and s'thitiardha. By 
successive approximation, we get the steady value 
of sthiti ardha etc. 



Moksa 




Figure 4 - Sparsa and moksa of pata 

Notes : In figure 4, fictitious joining of sun 
and moon have been shown. Kranti' s are equal in 
a pata, but they may be in different direction. They 
have been shown in same direction. Sun and moon 
are always in adjacent quadrants for pata as shown 
in figure 2, but they are shown at one place for 
explaining equality of kranti of different parts of 
sun and moon. At point O, kranti of centres of 
sun and moon are equal. Hence, it is mid time of 
pata. When centres of moon and sun are at Mi 
and Si, their kranti antara is Mi Si and discs touch 
each other at Ti. Kranti difference is moving along 
Mi OM2. At position M2, S2, the discs just touch 
at T 2 . 

Thus before Mi or after M2 position, the 
krantis of no part of sun and moon will be same 
and there will be no pata. Between these positions, 



Mahdpata Varnana 745 

some point of moon will have same kranti as some 
point of sun, hence it will be pata. 

It is clear that at sparsa time 

Kranti antara M1S1 = M1T1 + Ti Si 

= semi diameters of (sun + moon). 

Similarly at moksa time also it will be equal 
to manaikyardha. 

First we calculate the approximate position Si, 
S 2 from the kranti antara gati at O. Then we correct 
it with the gati at approximate positions of Si and 
S 2 to get more correct value. By repeating this 
process, we get the accurate value. 

This is only a diagram to explain equality of 
kranti of different points, there is no closeness of 
sun or moon as in ecliptic. 

Verses 43-45 : Effects of pata - According to 
Surya siddhanta the time of pata from beginning 
to end is fiery like burning fire and all auspicious 
works like marriages, sacred thread ceremony etc 
are prohibited during this. 

Pata arises due to equality of kranti' s of sun 
and moon. That destroys all results of noble deeds. 

By knowing the period of pata, penances like 
bath, charity, mantra, sraddha, worship, offers in 
fire all give good results, as on the occasion of 
eclipse. 

Verses 46-49 : Duration of pata 

Average duration of pata is two prahara or 
15 danda. Minimum duration is 9 danda and 

• • • • 

maximum duration is 2/20 days. 



74^ Siddhanta Darpana 

When ucca and pata of moon is in the last 
part of 12th rasi (mina), sayana sun is near mina 
and sayana moon is near kanya, then pata duration 
is minimum. 

When candra, its ucca, and pata all are at 
beginning of karka rasi, sayana sun is at the end 
of dhanu rasi, then pata is for maximum duration. 
If pata of candra is between mithuna 28° to 
karka 1 ° instead of karka beginning, then there is 
minimum time between two equalities of krantis 
(i.e. mahapatas). If at the end of ayana, spasta 
kranti of moon is more than 28°, then krantis 
cannot be equal. 

Notes : (1) Here pata has been used as short 
form of two mahapatas - Vaidhrti and vyatipata - 
when kranti of sun and moon are equal. But pata 
means the point of inter section of a planets orbit 
with ecliptic which is sun's orbit. For moon's orbit, 
pata's are called rahu or katu. As in case of all 
orbits, the pata point after which planet starts 
having north sara, the ascending node (rahu) is 
pata of moon. 

When moon, its ucca and its pata are in 
beginning of karka i.e. 90°, then sun in 270° will 
cause vaidhrti. Then moon has almost zero sara 
and its true north kranti is equal to sun kranti in 
opposite south direction. Speed of krantis will be 
slowest and speed of moon also will be slowest 
near its ucca, hence its pata will be longest. 

If moon pata is between mithuna 28° to karka 
1° (i.e. 88° to 91°) then within this movement of 
3° pata, just before pata position moon kranti will 



Mahapdta Varnana 747 

be less then 23-1/2° equal to kranti of sun before 
270°. After pata, sara of moon will rapidly increase 
and spasta kranti will be equal to sun kranti 
maximum at 270°. Hence next pata will come 
earliest. 

Opposite to the longest pata, moon at 180° 
and sun at 360° (vyati pata), if ucca and pata of 
moon are near 360/ then speed of moon is 
maximum, 0° kranti period will be for lowest period 
as kranti speed is maximum at 0° kranti and sara. 
hence pata is of smallest duration. 

For such situations, maximum and minimum 
periods of patas have already been given. 

(2) Maximum kranti of sun can be only upto 
23-1/2° in either direction. However, due to sara, 
moon can have kranti upto 28-1/2° due to its 
parama sara of 5°, when madhya kranti and sara 
both are maximum and in same direction. Then 
moon's kranti will be between 23-1/2 to 28-1/2° and 
sun's kranti will be always less than 23-1/2°. Hence 
true krantis can not be equal and there can be ng 
true pata, though madhya pata will occur. 

Verses 50-54 : Gola and ayana for pata 

For calculating true pata, sara of moon 
changes due to its orbit (distance from its pata 
rahu). But madhya kranti is same as kranti of sun 
in that ayana. At gola sandhi (zero sara) sphuta 
kranti doesn't change due to sara. But in ayana 
sandhi (maximum kranti but least kranti speed), 
kranti gati changes due to sara gati. Reason is that 
kranti gati is more in gola sandhi (at equator) and 
least in ayana sandhi (maximum kranti position). 



748 Siddhanta Darpana 

• 

In south and north gola, north south motion 
of moon due to sara doesn't change its total kranti 
gati. Being deflected north or south due to pata, 
moon still continues its motion on kranti vrtta. It 
is not affected, whatever may be the value of sara. 

Varahamihira has described gola and ayana 
system for mahapatas very logically in his 
Brhatyatra book. 

Notes : This is an objective description and 
needs no further comment. Brhatyatra is not a well 
known book of Varahamihira who has written three 
texts in three branches of jyotisa - Brhatsarhhita 
(Sarhhita), Brhatjataka (astrology - phalita) and 
Panca siddhantika (astronomy). 

Verses 55-58 : Inauspicious times 

In grahasphuta - chapter 5, 27 yogas have 
been described according to sum of rasi etc of sun 
and moon. Out of them 27th yoga is vaidhrti and 
17th is vyati pata. These yogas are very fiery 
because sun and moon become very angry, their 
aspects being inclined at same angle to equator, in 
same manner as two bullocks become angry when 
they are forced to move together round a pole for 
crushing oil seeds or separating grain chaff. 

From Surya siddhanta - Last quarters of aslesa, 
jyestha and revati - rasi and naksatra both have 
their borders. Hence last quarters (l/4th part) of 
these naksatras is called ganda. Half of first quarter 
(first l/8th part) of^next naksatras (magna, mula 
and asvinl) are called gandanta. 

All auspicious works are prohibited in sandhi 
(junction) of rasis. Last navamsa of karka, vrscika 



Mahapata Varnana 749 

and mlna rasi are in mlna rasi. First navamsa of 
next rasis (sirhha, dhanu and mesa) falls in mesa 
rasi. Hence all these navamsa are also bad. Like 
gandanta, these navamsa also fall in the junction 
of rasi and naksatra, hence good works are 
prohibited in them. Visti (bhadra) etc bad karanas 
are also to be avoided. 

Notes : This has nothing to do with ganita 
jyotisa. This can be considered use of these 
calculations of pata, naksatra karana and yoga. 

Surya siddhanta explains that 3 vyati patas, 
3 rasi sandhi and 3 naksatra sandhi all are very 
bad. 

Here 3 types of vyatipata are - mahapata called 
vyatipata and vaidhrti, yogas named vyati pata and 
vaidhrti. Mahapata are of two types - one from 
mean value of kranti and one from true kranti, 
hence three types of vyatipatas. 

12 rasis or 27 naksatras both are equal to 360° 
or full circle. Hence 1 rasi is equal to 2-1/4 = 9/4 
naksatras. Thus when 4 rasis are complete, 9 
naksatras also are completed, and their junctions 
combine. 

To tally rasi with naksatra, each naksatra is 
divided into 4 quarters, so that each rasi has 9 
quarters. Each rasi is also divided into 9 parts 
called navamsa. Thus, 1 navamsa = 1 quarter 
naksatra = 3° 20'. Navamsa also is counted like rasi 
starting with 1st navamsa of mesa as mesa, 2nd 
navamsa as vrsa etc. 

• » • 

Thus at the and of 4, 8, 12 rasis, 9th, 18th 
and 27th naksatras i.e. mlna navamsa is completed. 
Next navamsa i.e. 1st navamsa of 5, 9 1st rasis are 



750 Siddhdnta Darpana 

mesa navamsa. According to rules stated, last 
quarter of 9th, 18th and 27th rasis or first half 
quarters of next naksatras are bad. If a child is 
born during this period (i.e. if moon is in ganda 
or gandanta naksatra), that naksatra is worshipped 
when it comes again (on 27th day of birth). 

As the seventh day Sunday was not meant 
for work in Christianity, 7th karana visti is not 
good for starting any important work or for 
proceeding on a journey. It is also called bhadra 
(meaning good - probably for holiday purpose). 

Verses 59-62 : Comments on the siddhanta 
methods - Brahma took 47,400 divine years in 
creation of world, which is called srsti kala (creation 
period). From next day after creation, revolutions 
of graha, their ucca and pata etc started. Hence it 
has aheady been stated that for calculation of graha 
etc, the years of creation will be deducted from 
the years counted from beginning of kalpa. 

After completion of creation, caitra sukla 
pratipada was the first tithi. Then sun was rising 
in Yamakotipattana and it was mid night in Lanka. 
This day was named as ravivara (sunday). From 
that instant Brahma left graha, ucca and pata to 
move in their orbits from first point of asvini 
naksatra (mesa 0°) From that time only days, 
months, years, kranti and revolutions of graha etc 
started. They had not started from start of day of 
Brahma (called kalpa). From that time, only ghati 
.(1/60 of a civil day), yuga and manu etc started. 

Sages like Parasara have described king, 
ministers and protectors of the years, clouds like 
drona and puskara, rulers of grains etc, parts of 



Mahapata Varnana 751 

fire, rain and deceases, raja yoga etc for predicting 
good or bad results of future. Sometimes, they 
give the said results, sometimes they don't. Due 
to that, these have not been described here, as in 
other siddhanta texts. 

Sun and moon complete their revolutions at 
the end of every yuga and also in l/4th part of 
yuga. During a quarter of yuga (10,80,000 years), 
savana ahargana (civil days) are (39,44,79,457). At 
the end of dvapara, srstyabda (years since creation 
end) was (1,95,58,80,000). This divided by years of 
a quarter yuga (10,80,000) gives quotient (1811) and 
zero remainder, hence there is no need to state 
dhruva (positions) of sun and moon at the end of 
dvapara (after complete revolutions they are again 
at start of mesa 0°). 

Verses 63-66 : Start of Karana for this book 

From beginning of creation to dvapara end, 
past years (years completed at entry of mean sun 
in mesa) were (1, 95, 58, 80, 000), and at (4970) 
completed years in kali (1869 AD - Karanabda) the 
aharganas are (7, 14, 40, 22, 96, 627) arid (18, 15, 
334) from creation and kali. Both are correct as 
checked by vara (weekdays). 

At beginning of karanabda, when mean sun 
had entered mesa, first day according to mean 
value (sun and moon) was caitra sukla pratipada. 
The dhruva stated for that day (mean positions at 
beginning of year), when added to daily motion 
for lapsed days, becomes dhruva of madhyama 
graha acording to surya siddhanta. Ahargana of 
karanabda starts with tuesday (mangala vara). 



752 Siddhanta Darpana 

The day before beginning of karanabda has 
been assumed monday. That day was caitra sukla 
pratipada. (mean speed). According to spasta 
position it was vaisakha adhimasa (extra month) 
pratipada. Hence the day before start of karanabda 
is correct caitra pratipada according to mean speed 
and monday, which is convenient day for stating 
dhruvas. 

From starting point of karanabda, (18, 15, 
334-15) days before, kaliyuga had started at mid 
night at Lanka. According to ancient authorities, 
that was caitra sukla pratipada by mean positions. 
Again first day of karanabda is in vaisakha by true 
position. To find this caitra sukla pratipada, dhruva 
at t